Maharashtra Board Class 10 Maths Solutions Chapter 4 Financial Planning Problem Set 4B

Maharashtra State Board Class 10 Maths Solutions Chapter 4 Financial Planning Problem Set 4B

Financial Planning Class 10 Problem Set 4b Question 1. Write the correct alternative for the following questions.

i. If the Face Value of a share is ₹ 100 and Market value is ₹ 75, then which of the following statement is correct?
(A) The share is at premium of ₹ 175
(B) The share is at discount of ₹ 25
(C) The share is at premium of ₹ 25
(D) The share is at discount of ₹ 75
Answer:
(B)

ii. What is the amount of dividend received per share of face value ₹ 10 if dividend declared is 50%.
(A) ₹ 50
(B) ₹ 5
(C) ₹ 500
(D) ₹ 100
Answer:
Dividend = 10 × \(\frac { 50 }{ 100 } \) = ₹ 5
(B)

iii. The NAV of a unit in mutual fund scheme is ₹ 10.65, then find the amount required to buy 500 such units.
(A) 5325
(B) 5235
(C) 532500
(D) 53250
Answer:
(A)

iv. Rate of GST on brokerage is _______
(A) 5%
(B) 12%
(C) 18%
(D) 28%
Answer:
(C)

v. To find the cost of one share at the time of buying the amount of Brokerage and GST is to be ______ MV of share.
(A) added to
(B) subtracted from
(C) Multiplied with
(D) divided by
Answer:
(A)

Problem Set 4b Algebra Class 10 Question 2. Find the purchase price of a share of FV ₹ 100 if it is at premium of ₹ 30. The brokerage rate is 0.3%.
Solution:
Here, Face Value of share = ₹ 100,
premium = ₹ 30, brokerage = 0.3%
MV = FV + Premium
= 100 + 30
= ₹ 130
Brokerage = 0.3% of MV
= \(\frac { 0.3 }{ 100 } \) × 130 = ₹ 0.39
Purchase price of a share = MV + Brokerage
= 130 + 0.39
= ₹ 130.39
Purchase price of a share is ₹ 130.39.

Question 3.
Prashant bought 50 shares of FV ₹ 100, having MV ₹ 180. Company gave 40% dividend on the shares. Find the rate of return on investment.
Solution:
Here, Number of shares = 50, FV = ₹ 100,
MV = ₹ 180, rate of dividend = 40%
∴ Sum invested = Number of shares × MV
= 50 × 180
= ₹ 9000
Dividend per share = 40% of FV
= \(\frac { 40 }{ 100 } \) × 100
Dividend = ₹ 40
∴ Total dividend on 50 shares = 50 × 40
= ₹ 2000
Maharashtra Board Class 10 Maths Solutions Chapter 4 Financial Planning Problem Set 4B 1
∴ Rate of return on investment is 22.2%.

Question 4.
Find the amount received when 300 shares of FV ₹ 100, were sold at a discount of ₹ 30.
Solution:
Here, FV = ₹ 100, number of shares = 300,
discount = ₹ 30
MV of 1 share = FV – Discount
= 100 – 30 = ₹ 70
∴ MV of 300 shares = 300 × 70
= ₹ 21,000
∴ Amount received is ₹ 21,000.

Question 5.
Find the number of shares received when ₹ 60,000 was invested in the shares of FV ₹ 100 and MV ₹ 120.
Solution:
Here, FV = ₹ 100, MV = ₹ 120,
Sum invested = ₹ 60,000
Maharashtra Board Class 10 Maths Solutions Chapter 4 Financial Planning Problem Set 4B 2
∴ Number of shares received were 500.

Question 6.
Smt. Mita Agrawal invested ₹ 10,200 when MV of the share is ₹ 100. She sold 60 shares when the MV was ₹ 125 and sold remaining shares when the MV was ₹ 90. She paid 0.1% brokerage for each trading. Find whether she made profit or loss? and how much?
Solution:
For purchasing shares:
Here, sum invested = ₹ 10,200, MV = ₹ 100
Maharashtra Board Class 10 Maths Solutions Chapter 4 Financial Planning Problem Set 4B 3
For selling shares:
60 shares sold at MV of ₹ 125.
∴ MV of 60 shares = 125 × 60
= ₹ 7500
Brokerage = \(\frac { 0.1 }{ 100 } \) × 7500 = ₹ 7.5
∴ Sale value of 60 shares = 7500 – 7.5 = ₹ 7492.5
Now, remaining shares = 102 – 60 = 42
But 42 shares sold at MV of ₹ 90.
∴ MV of 42 shares = 42 × 90 = ₹ 3780
∴ Brokerage = \(\frac { 0.1 }{ 100 } \) × 3780 = ₹ 3.78
∴ Sale value of 42 shares = 3780 – 3.78 = ₹ 3776.22
Total sale value = 7492.5 + 3776.22 = ₹ 11268.72
Since, Purchase value < Sale value
∴ Profit is gained.
∴ Profit = Sale value – Purchase value
= 11268.72 – 10210.2
= ₹ 1058.52
∴ Smt. Mita Agrawal gained a profit of ₹ 1058.52.

Question 7. Market value of shares and dividend declared by the two companies is given below.
Face value is same and it is 7 100 for both the shares. Investment in which company is more profitable?
i. Company A – ₹ 132,12%
ii Company B – ₹ 144,16%
Solution:
For company A:
FV = ₹ 100, MV = ₹ 132,
Rate of dividend = 12%
Dividend = 12% of FV
Maharashtra Board Class 10 Maths Solutions Chapter 4 Financial Planning Problem Set 4B 4
∴ Rate of return of company B is more.
∴ Investment in company B is more profitable.

Question 8. Shri. Aditya Sanghavi invested ₹ 50,118 in shares of FV ₹ 100, when the market value is ₹ 50. Rate of brokerage is 0.2% and Rate of GST on brokerage is 18%, then How many shares were purchased for ₹ 50,118?
Solution:
Here, FV = ₹ 100, MV = ₹ 50
Purchase value of shares = ₹ 50118,
Rate of brokerage = 0.2%, Rate of GST = 18%
Brokerage = 0.2% of MV
Maharashtra Board Class 10 Maths Solutions Chapter 4 Financial Planning Problem Set 4B 5
Maharashtra Board Class 10 Maths Solutions Chapter 4 Financial Planning Problem Set 4B 6
∴ 1000 shares were purchased for ₹ 50,118.

Question 9. Shri. Batliwala sold shares of ₹ 30,350 and purchased shares of ₹ 69,650 in a day. He paid brokerage at the rate of 0.1% on sale and purchase. 18% GST was charged on brokerage. Find his total expenditure on brokerage and tax.
Solution:
Total amount = sale value + Purchase value
= 30350 + 69650
= ₹ 1,00,000
Rate of Brokerage = 0.1 %
Brokerage = 0.1 % of 1,00,000
= \(\frac { 0.1 }{ 100 } \) × 1,00,000
= ₹ 100
Rate of GST = 18%
∴ GST = 18 % of brokerage
= \(\frac { 18 }{ 100 } \) × 100
∴ GST = ₹ 18
Total expenditure on brokerage and tax
= 100 + 18 = ₹ 118
∴ Total expenditure on brokerage and tax is ₹ 118.

Alternate Method:
Brokerage = 0.1 %, GST = 18%
At the time of selling shares:
Total sale amount of shares = ₹ 30,350
Brokerage = 0.1% of 30,350
Maharashtra Board Class 10 Maths Solutions Chapter 4 Financial Planning Problem Set 4B 7
For purchasing shares:
Total purchase amount of shares = ₹ 69,650
Brokerage = 0.1% of 69,650
= \(\frac { 0.1 }{ 100 } \) × 69650
= ₹ 69.65
GST = 18% of 69.65
= \(\frac { 18 }{ 100 } \) × 69.65
= ₹ 12.537
∴ Total expenditure on brokerage and tax = Brokerage and tax on selling + Brokerage and tax on purchasing
= (30.35 + 5.463) + (69.65 + 12.537)
= ₹ 118
∴ Total expenditure on brokerage and tax is ₹ 118.

Question 10. Sint. Aruna Thakkar purchased 100 shares of FV 100 when the MV is ₹ 1200. She paid brokerage at the rate of 0.3% and 18% GST on brokerage. Find the following –
i. Net amount paid for 100 shares.
ii. Brokerage paid on sum invested.
iii. GST paid on brokerage.
iv. Total amount paid for 100 shares.
Solution:
Here, FV = ₹ 100,
Number of shares = 100, MV = ₹ 1200
Brokerage = 0.3%, GST = 18%
i. Sum invested = Number of shares × MV
= 100 × 1200 = ₹ 1,20,000
∴ Net amount paid for 100 shares is ₹ 1,20,000.

ii. Brokerage = 0.3% of sum invested
= \(\frac { 0.3 }{ 100 } \) × 1,20,000 = ₹ 360
∴ Brokerage paid on sum invested is ₹ 360.

iii. GST = 18% of brokerage
= \(\frac { 18 }{ 100 } \) × 360 = ₹ 64.80
∴ GST paid on brokerage is ₹ 64.80.

iv. Total amount paid for 100 shares
= Sum invested + Brokerage + GST
= 1,20,000 + 360 + 64.80
= ₹ 1,20,424.80
∴ Total amount paid for 100 shares is ₹ 1,20,424.80.

Question 11. Smt. Anagha Doshi purchased 22 shares of FV ₹ 100 for Market Value of ₹ 660. Find the sum invested. After taking 20% dividend, she sold all the shares when market value was ₹ 650. She paid 0.1% brokerage for each trading done. Find the percent of profit or loss in the share trading. (Write your answer to the nearest integer)
Solution:
For purchasing shares:
Here, FV = ₹ 100, MV = ₹ 660, Number of shares = 22, rate of brokerage = 0.1%
Sum invested = MV × Number of shares
= 660 × 22
= ₹ 14,520
Brokerage = 0.1 % of sum invested
= \(\frac { 0.1 }{ 100 } \) × 14520 = ₹ 14.52
∴ Amount invested for 22 shares
= Sum invested + Brokerage
= 14520 + 14.52
= ₹ 14534.52
For dividend:
Rate of dividend = 20%
∴ Dividend per share = 20 % of FV
Maharashtra Board Class 10 Maths Solutions Chapter 4 Financial Planning Problem Set 4B 8
∴ Percentage of profit in the share trading is 1 % (nearest integer).

Alternate Method:
For purchasing share:
Here, FV = ₹ 100, MV = ₹ 660, Number of shares = 22, rate of brokerage = 0.1%
Sum invested = MV × Number of shares
= 660 × 22
= ₹ 14,520
Brokerage = 0.1 % of MV
= \(\frac { 0.1 }{ 100 } \) × 660 = ₹ 0.66
Amount invested for 1 share = 660 + 0.66
= ₹ 660.66
For dividend:
Rate of dividend = 20%
Dividend = 20% of FV = \(\frac { 20 }{ 100 } \) × 100 = ₹ 20
For selling share:
MV = ₹ 650, rate of brokerage = 0.1%
Brokerage = 0.1 % of MV
= \(\frac { 0.1 }{ 100 } \) × 650 = ₹ 0.65 100
Amount received after selling 1 share
= 650 – 0.65 = 649.35
∴ Amount received including divided
= selling price of 1 share + dividend per share
= 649.35 + 20
= ₹ 669.35
Since, income > Amount invested
∴ Profit is gained.
∴ profit = 669.35 – 660.66 = ₹ 8.69
Profit Percentage = \(\frac { 8.69 }{ 660.66 } \) × 100= 1.31%
∴ Percentage of profit in the share trading is 1 % (nearest integer).

Maharashtra Board Class 10 Maths Solutions

Maharashtra Board Class 9 Maths Solutions Chapter 4 Ratio and Proportion Practice Set 4.2

Maharashtra Board Class 9 Maths Solutions Chapter 4 Ratio and Proportion Practice Set 4.2

Question 1.
Using the property \(\frac { a }{ b }\) = \(\frac { ak }{ bk }\), fill in the blanks by substituting proper numbers in the following.
Maharashtra Board Class 9 Maths Solutions Chapter 4 Ratio and Proportion Practice Set 4.2 1
Solution:
Maharashtra Board Class 9 Maths Solutions Chapter 4 Ratio and Proportion Practice Set 4.2 2
Maharashtra Board Class 9 Maths Solutions Chapter 4 Ratio and Proportion Practice Set 4.2 3

Question 2.
Find the following ratios.
i. The ratio of radius to circumference of the circle.
ii. The ratio of circumference of circle with radius r to its area.
iii. The ratio of diagonal of a square to its side, if the length of side is 7 cm.
iv. The lengths of sides of a rectangle are 5 cm and 3.5 cm. Find the ratio of numbers denoting its perimeter to area.
Solution:
i. Let the radius of circle be r.
then, its circumference = 2πr
Ratio of radius to circumference of the circle
Maharashtra Board Class 9 Maths Solutions Chapter 4 Ratio and Proportion Practice Set 4.2 4
The ratio of radius to circumference of the circle is 1 : 2π.

ii. Let the radius of the circle is r.
∴ circumference = 2πr and area = πr2
Ratio of circumference to the area of circle
Maharashtra Board Class 9 Maths Solutions Chapter 4 Ratio and Proportion Practice Set 4.2 5
∴ The ratio of circumference of circle with radius r to its area is 2 : r.

iii. Length of side of square = 7 cm
∴ Diagonal of square = √2 x side
= √2 x 7
= 7 √2 cm
Ratio of diagonal of a square to its side
Maharashtra Board Class 9 Maths Solutions Chapter 4 Ratio and Proportion Practice Set 4.2 6
∴ The ratio of diagonal of a square to its side is √2 : 1.

iv. Length of rectangle = (l) = 5 cm,
Breadth of rectangle = (b) = 3.5 cm
Perimeter of the rectangle = 2(l + b)
= 2(5 + 3.5)
= 2 x 8.5
= 17 cm
Area of the rectangle = l x b
= 5 x 3.5
= 17.5 cm2
Ratio of numbers denoting perimeter to the area of rectangle
Maharashtra Board Class 9 Maths Solutions Chapter 4 Ratio and Proportion Practice Set 4.2 7
∴ Ratio of numbers denoting perimeter to the area of rectangle is 34 : 35.

Question 3.
Compare the following
Maharashtra Board Class 9 Maths Solutions Chapter 4 Ratio and Proportion Practice Set 4.2 8
Solution:
Maharashtra Board Class 9 Maths Solutions Chapter 4 Ratio and Proportion Practice Set 4.2 9
Maharashtra Board Class 9 Maths Solutions Chapter 4 Ratio and Proportion Practice Set 4.2 10
Maharashtra Board Class 9 Maths Solutions Chapter 4 Ratio and Proportion Practice Set 4.2 11
Maharashtra Board Class 9 Maths Solutions Chapter 4 Ratio and Proportion Practice Set 4.2 12

Question 4.
Solve.
ABCD is a parallelogram. The ratio of ∠A and ∠B of this parallelogram is 5 : 4. FInd the measure of ∠B. [2 Marksl
Solution:
Ratio of ∠A and ∠B for given parallelogram is 5 : 4
Let the common multiple be x.
Maharashtra Board Class 9 Maths Solutions Chapter 4 Ratio and Proportion Practice Set 4.2 13
m∠A = 5x°and m∠B=4x°
Now, m∠A + m∠B = 180° …[Adjacent angles of a parallelogram arc supplementary]
∴ 5x° + 4x°= 180°
∴ 9x° = 180°
∴ x° = 20°
∴ m∠B=4x°= 4 x 20° = 80°
∴ The measure of ∠B is 800.

ii. The ratio of present ages of Albert and Salim is 5 : 9. Five years hence ratio of their ages will be 3 : 5. Find their present ages.
Solution:
The ratio of present ages of Albert and Salim is 5 : 9
Let the common multiple be x.
∴ Present age of Albert = 5x years and
Present age of Salim = 9x years
After 5 years,
Albert’s age = (5x + 5) years and
Salim’s age = (9x + 5) years
According to the given condition,
Five years hence ratio of their ages will be 3 : 5
\(\frac{5 x+5}{9 x+5}=\frac{3}{5}\)
∴ 5(5x + 5) = 3(9x + 5)
∴ 25x + 25 = 27x + 15
∴ 25 – 15 = 27 x – 25 x
∴ 10 = 2x
∴ x = 5
∴ Present age of Albert = 5x = 5 x 5 = 25 years
Present age of Salim = 9x = 9 x 5 = 45 years
∴ The present ages of Albert and Salim are 25 years and 45 years respectively.

iii. The ratio of length and breadth of a rectangle is 3 : 1, and its perimeter is 36 cm. Find the length and breadth of the rectangle.
Solution:
The ratio of length and breadth of a rectangle is 3 : 1
Let the common multiple be x.
Length of the rectangle (l) = 3x cm
and Breadth of the rectangle (b) = x cm
Given, perimeter of the rectangle = 36 cm
Since, Perimeter of the rectangle = 2(l + b)
∴ 36 = 2(3x + x)
∴ 36 = 2(4x)
∴ 36 = 8x
∴ \(x=\frac{36}{8}=\frac{9}{2}=4.5\)
Length of the rectangle = 3x = 3 x 4.5 = 13.5 cm
∴ The length of the rectangle is 13.5 cm and its breadth is 4.5 cm.

iv. The ratio of two numbers is 31 : 23 and their sum is 216. Find these numbers.
Solution:
The ratio of two numbers is 31 : 23
Let the common multiple be x.
∴ First number = 31x and
Second number = 23x
According to the given condition,
Sum of the numbers is 216
∴ 31x + 23x = 216
∴ 54x = 216
∴ x = 4
∴ First number = 31x = 31 x 4 = 124
Second number = 23x = 23 x 4 = 92
∴ The two numbers are 124 and 92.

v. If the product of two numbers is 360 and their ratio is 10 : 9, then find the numbers.
Solution:
Ratio of two numbers is 10 : 9
Let the common multiple be x.
∴ First number = 10x and
Second number = 9x
According to the given condition,
Product of two numbers is 360
∴ (10x) (9x) = 360
∴ 90x2 = 360
∴ x2 = 4
∴ x = 2 …. [Taking positive square root on both sides]
∴ First number = 10x = 10x2 = 20
Second number = 9x = 9x2 = 18
∴ The two numbers are 20 and 18.

Question 5.
If a : b = 3 : 1 and b : c = 5 : 1, then find the value of [3 Marks each]
Maharashtra Board Class 9 Maths Solutions Chapter 4 Ratio and Proportion Practice Set 4.2 14
Solution:
Given, a : b = 3 : 1
∴ \(\frac { a }{ b }\) = \(\frac { 3 }{ 1 }\)
∴ a = 3b ….(i)
and b : c = 5 : 1
∴ \(\frac { b }{ c }\) = \(\frac { 5 }{ 1 }\)
b = 5c …..(ii)
Substituting (ii) in (i),
we get a = 3(5c)
∴ a = 15c …(iii)
Maharashtra Board Class 9 Maths Solutions Chapter 4 Ratio and Proportion Practice Set 4.2 15

Ratio and Proportion 9th Class Practice Set 4.1 Question 6. If \(\sqrt{0.04 \times 0.4 \times a}=0.4 \times 0.04 \times \sqrt{b}\) , then find the ratio \(\frac { a }{ b }\).
Solution:
\(\sqrt{0.04 \times 0.4 \times a}=0.4 \times 0.04 \times \sqrt{b}\) … [Given]
∴ 0.04 x 0.4 x a = (0.4)2 x (0.04)2 x b … [Squaring both sides]
Maharashtra Board Class 9 Maths Solutions Chapter 4 Ratio and Proportion Practice Set 4.2 16

9th Algebra Practice Set 4.2 Question 7. (x + 3) : (x + 11) = (x – 2) : (x + 1), then find the value of x.
Solution:
(x + 3) : (x + 11) = (x- 2) : (x+ 1)
\(\quad \frac{x+3}{x+11}=\frac{x-2}{x+1}\)
∴ (x + 3)(x +1) = (x – 2)(x + 11)
∴ x(x +1) + 3(x + 1) = x(x + 11) – 2(x + 11)
∴ x2 + x + 3x + 3 = x2 + 1 lx – 2x – 22
∴ x2 + 4x + 3 = x2 + 9x – 22
∴ 4x + 3 = 9x – 22
∴ 3 + 22 = 9x – 4x
∴ 25 = 5x
∴ x = 5

Maharashtra Board Class 9 Maths Solutions

Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.4

Maharashtra State Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.4

Practice Set 7.4 Geometry Class 10 Question 1. In the adjoining figure, A is the centre of the circle. ∠ABC = 45° and AC = 7\(\sqrt { 2 }\) cm. Find the area of segment BXC, (π = 3.14)
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.4 1
Solution:
In ∆ABC,
AC = AB … [Radii of same circle]
∴ ∠ABC = ∠ACB …[Isosceles triangle theorem]
∴ ∠ABC = ∠ACB = 45°
In ∆ABC,
∠ABC + ∠ACB + ∠BAC = 180° … [Sum of the measures of angles of a triangle is 180° ]
∴ 45° + 45° + ∠BAC = 180°
∴ 90° + ∠BAC = 180°
∴ ∠BAC = 90°
Let ∠BAC = θ = 90°
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.4 2
∴ The area of segment BXC is 27.93 cm2.

10th Class Geometry Practice Set 7.4 Question 2. In the adjoining figure, O is the centre of the circle.
m(arc PQR) = 60°, OP = 10 cm. Find the area of the shaded region.
(π = 3.14, \(\sqrt { 3 }\) = 1.73)
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.4 3
Given: m(arc PQR) = 60°, radius (r) = OP = 10 cm
To find: Area of shaded region.
Solution:
∠POR = m (arc PQR) …[Measure of central angle]
∴ ∠POR = θ = 60°
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.4 4
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.4 5
∴ The area of the shaded region is 9.08 cm2.

7.4 Class 10 Question 3. In the adjoining figure, if A is the centre of the circle, ∠PAR = 30°, AP = 7.5, find the area of the segment PQR. (π = 3.14)
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.4 6
Given: Central angle (θ) = ∠PAR = 30°,
radius (r) = AP = 7.5
To find: Area of segment PQR.
Solution:
Let ∠PAR = θ = 30°
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.4 7
∴ The area of segment PQR is 0.65625 sq. units.

Chapter 7 Maths Class 10 Question 4. In the adjoining figure, if O is the centre of the circle, PQ is a chord, ∠POQ = 90°, area of shaded region is 114 cm2, find the radius of the circle, (π = 3.14)
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.4 8
Given: Central angle (θ) = ∠POQ= 90°,
A (segment PRQ) = 114 cm2
To find: Radius (r).
Solution:
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.4 9
…[Taking square root of both sides]
∴ r = 20 cm
∴ The radius of the circle is 20 cm.

Mensuration Questions for Class 10 Question 5. A chord PQ of a circle with radius 15 cm subtends an angle of 60° with the centre of the circle. Find the area of the minor as well as the major segment. (π = 3.14, \(\sqrt { 3 }\) = 1.73)
Given: Radius (r) =15 cm, central angle (θ) = 60°
To find: Areas of major and minor segments.
Solution:
Let chord PQ subtend ∠POQ = 60° at centre.
∴ θ = 60°
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.4 10
= 225 [0.0908]
= 20.43 cm2
∴ area of minor segment = 20.43 cm2
Area of circle = πr2
= 3.14 × 15 × 15
= 3.14 × 225
= 706.5 cm2
Area of major segment
= Area of circle – area of minor segment
= 706.5 – 20.43
= 686.07 cm2
Area of major segment 686.07 cm2
∴ The area of minor segment Is 20.43 cm2 and the area of major segment is 686.07 cm2.

Maharashtra Board Class 10 Maths Solutions

Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Problem Set 7

Maharashtra State Board Class 10 Maths Solutions Chapter 7 Mensuration Problem Set 7

Problem Set 7 Question 1. Choose the correct alternative answer for each of the following questions.

i. The ratio of circumference and area of a circle is 2 : 7. Find its circumference.
(A) 14 π
(B) \(\frac{7}{\pi}\)
(C) 7π
(D) \(\frac{14}{\pi}\)
Answer:
Problem Set 7 Geometry 10th
(A)

ii. If measure of an arc of a circle is 160° and its length is 44 cm, find the circumference of the circle.
(A) 66 cm
(B) 44 cm
(C) 160 cm
(D) 99 cm
Answer:

(D)

iii. Find the perimeter of a sector of a circle if its measure is 90° and radius is 7 cm.
(A) 44 cm
(B) 25 cm
(C) 36 cm
(D) 56 cm
Answer:

(B)

iv. Find the curved surface area of a cone of radius 7 cm and height 24 cm.
(A) 440 cm2
(B) 550 cm2
(C) 330 cm2
(D) 110 cm2
Answer:

(B)

v. The curved surface area of a cylinder is 440 cm2 and its radius is 5 cm. Find its height.
(A) \(\frac{44}{\pi}\) cm
(B) 22π cm
(C) 44π cm
(D) \(\frac{22}{\pi}\)
Answer:

(A)

vi. A cone was melted and cast into a cylinder of the same radius as that of the base of the cone. If the height of the cylinder is 5 cm, find the height of the cone.
(A) 15 cm
(B) 10 cm
(C) 18 cm
(D) 5 cm
Answer:
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Problem Set 7
(A)

vii. Find the volume of a cube of side 0.01 cm.
(A) 1 cm
(B) 0.001 cm3
(C) 0.0001 cm3
(D) 0.000001 cm3
Answer:
Volume of cube = (side)3
= (0.01)3 = 0.000001 cm3
(D)

viii. Find the side of a cube of volume 1 m3
(A) 1 cm
(B) 10 cm
(C) 100 cm
(D) 1000 cm
Answer:
Volume of cube = (side)3
∴ 1 = (side)3
∴ Side = 1 m
= 100 cm
(C)

Problem Set 7 Geometry Class 10 Question 2. A washing tub in the shape of a frustum of a cone has height 21 cm. The radii of the circular top and bottom are 20 cm and 15 cm respectively. What is the capacity of the tub? = (π = \(\frac { 22 }{ 7 } \))
Given: For the frustum shaped tub,
height (h) = 21 cm,
radii (r1) = 20 cm, and (r2) = 15 cm
To find: Capacity (volume) of the tub.
Solution:
Volume of frustum = \(\frac { 1 }{ 3 } \) πh (r12 + r22 + r1 × r2)
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Problem Set 7 7
∴ The capacity of the tub is 20.35 litres.

10th Geometry Problem Set 7 Question 3. Some plastic balls of radius 1 cm were melted and cast into a tube. The thickness, length and outer radius of the tube were 2 cm, 90 cm and 30 cm respectively. How many balls were melted to make the tube?
Given: For the cylindrical tube,
height (h) = 90 cm,
outer radius (R) = 30 cm,
thickness = 2 cm
For the plastic spherical ball,
radius (r1) = 1 cm
To find: Number of balls melted.
Solution:
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Problem Set 7 8
Inner radius of tube (r)
= outer radius – thickness of tube
= 30 – 2
= 28 cm
Volume of plastic required for the tube = Outer volume of tube – Inner volume of hollow tube
= πR2h – πr2h
= πh(R2 – r2)
= π × 90 (302 – 282)
= π × 90 (30 + 28) (30 – 28) …[∵ a2 – b2 = (a + b)(a – b)]
= 90 × 58 × 2π cm3
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Problem Set 7 9
∴ 7830 plastic balls were melted to make the tube.

Problem Set 7 Geometry Question 4.
A metal parallelopiped of measures 16 cm × 11cm × 10cm was melted to make coins. How many coins were made if the thickness and diameter of each coin was 2 mm and 2 cm respectively?
Given: For the parallelopiped.,
length (l) = 16 cm, breadth (b) = 11 cm,
height (h) = 10 cm
For the cylindrical coin,
thickness (H) = 2 mm,
diameter (D) 2 cm
To find: Number of coins made.
Solution:
Volume of parallelopiped = l × b × h
= 16 × 11 × 10
= 1760 cm3
Thickness of coin (H) = 2 mm
= 0.2 cm …[∵ 1 cm = 10 mm]
Diameter of coin (D) = 2 cm
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Problem Set 7 10
∴ 2800 coins were made by melting the parallelopiped.

Mensuration Problem Question 5.  The diameter and length of a roller is 120 cm and 84 cm respectively. To level the ground, 200 rotations of the roller are required. Find the expenditure to level the ground at the rate of ₹ 10 per sq.m.
Given: For the cylindrical roller,
diameter (d) =120 cm,
length = height (h) = 84 cm
To find: Expenditure of levelling the ground.
Solution:
Diameter of roller (d) = 120 cm
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Problem Set 7 11
Now, area of ground levelled in one rotation = curved surface area of roller
= 3.168 m2
∴ Area of ground levelled in 200 rotations
= 3.168 × 200 =
633.6 m2
Rate of levelling = ₹ 10 per m2
∴ Expenditure of levelling the ground
= 633.6 × 10 = ₹ 6336
∴ The expenditure of levelling the ground is ₹ 6336.

Question 6.
The diameter and thickness of a hollow metal sphere are 12 cm and 0.01 m respectively. The density of the metal is 8.88 gm per cm3. Find the outer surface area and mass of the sphere, [π = 3.14]
Given: For the hollow sphere,
diameter (D) =12 cm, thickness = 0.01 m
density of the metal = 8.88 gm per cm3
To find: i. Outer surface area of the sphere
ii. Mass of the sphere.
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Problem Set 7 12
Solution:
Diameter of the sphere (D)
= 12 cm
∴ Radius of sphere (R)
= \(\frac { d }{ 2 } \) = \(\frac { 12 }{ 2 } \) = 6 cm
∴ Surface area of sphere = 4πR2
= 4 × 3.14 × 62
= 452.16 cm2
Thickness of sphere = 0.01 m
= 0.01 × 100 cm …[∵ 1 m = 100 cm]
= 1 cm
∴ Inner radius of the sphere (r)
= Outer radius – thickness of sphere
= 6 – 1 = 5 cm
∴ Volume of hollow sphere
= Volume of outer sphere – Volume of inner sphere
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Problem Set 7 13
∴ The outer surface area and the mass of the sphere are 452.16 cm2 and 3383.19 gm respectively.

Question 7.
A cylindrical bucket of diameter 28 cm and height 20 cm was full of sand. When the sand in the bucket was poured on the ground, the sand got converted into a shape of a cone. If the height of the cone was 14 cm, what was the base area of the cone?
Given: For the cylindrical bucket,
diameter (d) = 28 cm, height (h) = 20 cm
For the conical heap of sand,
height (H) = 14 cm
To find: Base area of the cone (πR2).
Solution:
Diameter of the bucket (d) = 28 cm
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Problem Set 7 14
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Problem Set 7 15
The base area of the cone is 2640 cm2.

Question 8.
The radius of a metallic sphere is 9 cm. It was melted to make a wire of diameter 4 mm. Find the length of the wire.
Given: For metallic sphere,
radius (R) = 9 cm
For the cylindrical wire,
diameter (d) = 4 mm
To find: Length of wire (h).
Solution:
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Problem Set 7 16
∴ The length of the wire is 243 m.

Question 9.
The area of a sector of a circle of 6 cm radius is 157t sq.cm. Find the measure of the arc and length of the arc corresponding to the sector.
Given: Radius (r) = 6 cm,
area of sector = 15 π cm2
To find: i. Measure of the arc (θ),
ii. Length of the arc (l)
Solution:
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Problem Set 7 17
∴ The measure of the arc and the length of the arc are 150° and 5π cm respectively.

Question 10.
In the adjoining figure, seg AB is a chord of a circle with centre P. If PA = 8 cm and distance of chord AB from the centre P is 4 cm, find the area of the shaded portion.
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Problem Set 7
(π = 3.14, \(\sqrt { 3 }\) = 1.73)
Given: Radius (r) = PA = 8 cm,
PC = 4 cm
To find: Area of shaded region.
Solution:
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Problem Set 7 18
Similarly, we can show that, ∠BPC = 60°
∠APB = ∠APC + ∠BPC …[Angle sum property]
∴ θ = 60° + 60° = 120°
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Problem Set 7 19
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Problem Set 7 20
Area of shaded region = A(P-ADB) – A(∆APB)
= 66.98 – 27.68
= 39.30 cm2
∴ The area of the shaded region is 39.30 cm2.

Question 11.
In the adjoining figure, square ABCD is inscribed in the sector A-PCQ. The radius of sector C-BXD is 20 cm. Complete the following activity to find the area of shaded region.
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Problem Set 7 21
Solution:
Side of square ABCD
= radius of sector C-BXD = [20] cm
Area of square = (side)2 = 202 = 400 cm2 ….(i)
Area of shaded region inside the square = Area of square ABCD – Area of sector C-BXD
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Problem Set 7 22
Radius of bigger sector
= Length of diagonal of square ABCD
= \(\sqrt { 2 }\) × side
= 20 \(\sqrt { 2 }\) cm
Area of the shaded regions outside the square
= Area of sector A-PCQ – Area of square ABCD
= A(A – PCQ) – A(꠸ABCD)
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Problem Set 7 23
Alternate method:
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Problem Set 7 24
□ABCD is a square. … [Given]
Side of □ABCD = radius of sector (C-BXD)
= 20 cm
Radius of sector (A-PCQ) = Diagonal
= \(\sqrt { 2 }\) × side
= \(\sqrt { 2 }\) × 20
= 20 \(\sqrt { 2 }\) cm
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Problem Set 7 25
Now, Area of shaded region
= A(A-PCQ) – A(C-BXD)
= 628 – 314
= 314 cm2
∴ The area of the shaded region is 314 cm2.

Question 12.
In the adjoining figure, two circles with centres O and P are touching internally at point A. If BQ = 9, DE = 5, complete the following activity to find the radii of the circles.
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Problem Set 7 26
Solution:
Let the radius of the bigger circle be R and that of smaller circle be r.
OA, OB, OC and OD are the radii of the bigger circle.
∴ OA = OB = OC = OD = R
PQ = PA = r
OQ + BQ = OB … [B – Q – O]
OQ = OB – BQ = R – 9
OE + DE = OD ….[D – E – O]
OE = OD – DE = [R – 5]
As the chords QA and EF of the circle with centre P intersect in the interior of the circle, so by the property of internal division of two chords of a circle,
OQ × OA = OE × OF
∴ (R – 9) × R = (R – 5) × (R – 5) …[∵ OE = OF]
∴ R2 – 9R = R2 – 10R + 25
∴ -9R + 10R = 25
∴ R = [25units]
AQ = AB – BQ = 2r ….[B-Q-A]
∴ 2r = 50 – 9 = 41
∴ r = \(\frac { 41 }{ 2 } \) = 20.5 units

Maharashtra Board Class 10 Maths Solutions

Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.1

Maharashtra State Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.1

Practice Set 7.1 Geometry 10th Question 1. Find the volume of a cone if the radius of its base is 1.5 cm and its perpendicular height is 5 cm.
Given: For the cone,
radius (r) = 1.5 cm,
perpendicular height (h) = 5 cm
To find: Volume of the cone.
Solution:
Volume of cone = \(\frac { 1 }{ 3 } \) πr2h
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.1 1
∴ The volume of the cone is 11.79 cm3.

Mensuration Practice Set 7.1 Question 2. Find the volume of a sphere of diameter 6 cm. [π = 3.14]
Given: For the sphere, diameter (d) = 6 cm
To find: Volume of the sphere.
Solution:
Radius (r) = \(\frac { d }{ 2 } \) = \(\frac { 6 }{ 2 } \) = 3 cm
Volume of sphere = \(\frac { 4 }{ 3 } \) πr2
= \(\frac { 4 }{ 3 } \) × 3.14 × (3)3
= 4 × 3.14 × 3 × 3
= 113.04 cm3
∴ The volume of the sphere is 113.04 cm3.

Practice Set 7.1 Geometry Class 10 Question 3. Find the total surface area of a cylinder if the radius of its base is 5 cm and height is 40 cm. [π = 3.14]
Given: For the cylinder,
radius (r) = 5 cm,
height (h) = 40 cm
To find: Total surface area of the cylinder.
Solution:
Total surface area of cylinder = 2πr (r + h)
= 2 × 3.14 × 5 (5 + 40)
= 2 × 3.14 × 5 × 45
= 1413 cm2
The total surface area of the cylinder is 1413 cm2.

Practice Set 7.1 Geometry Question 4. Find the surface area of a sphere of radius 7 cm.
Given: For the sphere, radius (r) = 7 cm
To find: Surface area of the sphere.
Solution:
Surface area of sphere = Aπr2
= 4 × \(\frac { 22 }{ 7 } \) × (7)2
= 88 × 7
= 616 cm2
∴ The surface area of the sphere is 616 cm2.

Practice Set 7.1 Question 5. The dimensions of a cuboid are 44 cm, 21 cm, 12 cm. It is melted and a cone of height 24 cm is made. Find the radius of its base.
Given: For the cuboid,
length (l) = 44 cm, breadth (b) = 21 cm,
height (h) = 12 cm
For the cone, height (H) = 24 cm
To find: Radius of base of the cone (r).
Solution:
Volume of cuboid = l × b × h
= 44 × 21 × 12 cm3
Volume of cone = \(\frac { 1 }{ 3 } \) πr2H
= \(\frac { 1 }{ 3 } \) × \(\frac { 22 }{ 7 } \) × r2 × 24 cm3
Since the cuboid is melted to form a cone,
∴ volume of cuboid = volume of cone
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.1 2
∴ r2 = 21 × 21
∴ r = 21 cm …[Taking square root of both sides]
∴ The radius of the base of the cone is 21 cm.

10th Class Geometry Practice Set 7.1 Question 6. Observe the measures of pots in the given figures. How many jugs of water can the cylindrical pot hold?
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.1 3
Given: For the conical water jug,
radius (r) = 3.5 cm, height (h) = 10 cm
For the cylindrical water pot,
radius (R) = 7 cm, height (H) = 10 cm
To find: Number of jugs of water the cylindrical pot can hold.
Solution:
Volume of conical jug = \(\frac { 1 }{ 3 } \) πr2h
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.1 4
∴ The cylindrical pot can hold 12 jugs of water.

Mensuration Class 10 Practice Set 7.1 Question 7. A cylinder and a cone have equal bases. The height of the cylinder is 3 cm and the area of its base is 100 cm2. The cone is placed up on the cylinder. Volume of the solid figure so formed is 500 cm3. Find the total height of the figure
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.1 5
Given: For the cylindrical part,
height (h) = 3 cm,
area of the base (πr2)= 100 cm2
Volume of the entire figure = 500 cm3
To find: Total height of the figure.
Solution:
A cylinder and a cone have equal bases.
∴ they have equal radii.
radius of cylinder = radius of cone = r
Area of base = 100 cm2
∴ πr2 =100 …(i)
Let the height of the conical part be H.
Volume of the entire figure
= Volume of the entire + Volume of cone
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.1 6
∴ Height of conical part (H) =6 cm
Total height of the figure = h + H
= 3 + 6
= 9 cm
∴ The total height of the figure is 9 cm.

10th Geometry Practice Set 7.1 Question 8. In the given figure, a toy made from a hemisphere, a cylinder and a cone is shown. Find the total area of the toy.
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.1
Given: For the conical Part,
height (h) = 4 cm, radius (r) = 3 cm
For the cylindrical part,
height (H) = 40 cm, radius (r) = 3 cm
For the hemispherical part,
radius (r) = 3 cm
To find: Total area of the toy.
Solution:
Slant height of cone (l) = \(\sqrt{\mathrm{h}^{2}+\mathrm{r}^{2}}\)
= \(\sqrt{\mathrm{4}^{2}+\mathrm{3}^{2}}\)
= \(\sqrt { 16+9 }\)
= \(\sqrt { 25 }\) = 5 cm
Curved surface area of cone = πrl
= π × 3 × 5
= 15π cm2
Curved surface area of cylinder = 2πrH
= 2 × π × 3 × 40
= 240π cm2
Curved surface area of hemisphere = 2πr2
= 2 × π × 32
= 18π cm2
Total area of the toy
= Curved surface area of cone + Curved surface area of cylinder + Curved surface area of hemisphere
= 15π + 240π + 18π
= 2737 π cm2
∴ The total area of the toy is 273π cm2.

7.1.8 Practice Question 9. In the given figure, a cylindrical wrapper of flat tablets is shown. The radius of a tablet is 7 mm and its thickness is 5 mm. How many such tablets are wrapped in the wrapper?
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.1 7
Given: For the cylindrical tablets,
radius (r) = 7 mm,
thickness = height(h) = 5 mm
For the cylindrical wrapper,
diameter (D) = 14 mm, height (H) = 10 cm
To find: Number of tablets that can be wrapped.
Solution:
Radius of wrapper (R) = \(\frac { Diameter }{ 2 } \)
= \(\frac { 14 }{ 2 } \) = 7 mm
Height of wrapper (H) = 10 cm
= 10 × 10 mm
= 100 mm
Volume of a cylindrical wrapper = πR2H
= π(7)2 × 100
= 4900π mm3
Volume of a cylindrical tablet = πr2h
= π(7)2 × 5
= 245 π mm3
No. of tablets that can be wrapped
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.1 8
∴ 20 tables can be wrapped in the wrapper

Class 10 Maths 7.1 Question 10. The given figure shows a toy. Its lower part is a hemisphere and the upper part is a cone. Find the volume and the surface area of the toy from the measures shown in the figure.
(π = 3.14)
Given: For the conical part,
height (h) = 4 cm, radius (r) = 3 cm
For the hemispherical part,
radius (r) = 3 cm
To find: Volume and surface area of the toy.
Solution:
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.1 9
Now, volume of the toy
= Volume of cone + volume of hemisphere
= 12π + 18π
= 30π
= 30 × 3.14
= 94.20 cm3
Also, surface area of the toy
= Curved surface area of cone + Curved surface area of hemisphere
= 15π + 18π
= 33π
= 33 × 3.14
= 103.62 cm2
∴ The volume and surface area of the toy are 94.20 cm3 and 103.62 cm2 respectively.

Question 11.
Find the surface area and the volume of a beach ball shown in the figure.
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.1 10
Given: For the spherical ball,
diameter (d) = 42 cm
To find: Surface area and volume of the beach ball.
Solution:
Radius (r) = \(\frac { d }{ 2 } \) = \(\frac { 42 }{ 2 } \) = 21 cm
Surface area of sphere= 4πr2
= 4 × 3.14 × (21)2
= 4 × 3.14 × 21 × 21
= 5538.96 cm2
Volume of sphere = \(\frac { 4 }{ 3 } \) πr3
= \(\frac { 4 }{ 3 } \) × 3.14 × (21)3
= 4 × 3.14 × 7 × 21 × 21
= 38772.72 cm3
∴ The surface area and the volume of the beach ball are 5538.96 cm2 and 38772.72 cm3 respectively.

Question 12.
As shown in the figure, a cylindrical glass contains water. A metal sphere of diameter 2 cm is immersed in it. Find the volume of the water.
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.1 11
Given: For the metal sphere,
diameter (d) = 2 cm
For the cylindrical glass, diameter (D) =14 cm,
height of water in the glass (H) = 30 cm
To find: Volume of water in the glass.
Solution:
Let the radii of the sphere and glass be r and R respectively.
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.1 12
Volume of water with sphere in it = πR2H
= π × (7)2 × 30
= 1470π cm3
Volume of water in the glass
= Volume of water with sphere in it – Volume of sphere
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.1 13
∴ The volume of the water in the glass is 1468.67 π cm3 (i.e. 4615.80 cm3).

Maharashtra Board Class 10 Maths Chapter 7 Mensuration Intext Questions and Activities

Question 1.
The length, breadth and height of an oil can are 20 cm, 20 cm and 30 cm respectively as shown in the adjacent figure. How much oil will it contain? (1 litre = 1000 cm3) (Textbook pg. no.141)

Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.1
Given: For the cuboidal can,
length (l) = 20 cm,
breadth (b) = 20 cm,
height (h) = 30 cm
To find: Oil that can be contained in the can.
Solution:
Volume of cuboid = l × b × h
= 20 × 20 × 30
= 12000 cm3
= \(\frac { 12000 }{ 1000 } \) litres
= 12 litres
∴ The oil can will contain 12 litres of oil.

Question 2.
The adjoining figure shows the measures of a Joker’s dap. How much cloth is needed to make such a cap? (Textbook pg. no. 141)
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.1
Given: For the conical cap,
radius (r) = 10 cm,
slant height (l) = 21 cm
To find: Cloth required to make the cap.
Solution:
Cloth required to make the cap
= Curved surface area of the conical cap
= πrl = \(\frac { 22 }{ 7 } \) × 10 × 21
=22 × 10 × 3
= 660 cm2
∴ 660 cm2 of cloth will be required to make the cap.

Question 3.
As shown in the adjacent figure, a sphere is placed in a cylinder. It touches the top, bottom and the curved surface of the cylinder. If radius of the base of the cylinder is ‘r’,
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.1
i. what is the ratio of the radii of the sphere and the cylinder ?
ii. what is the ratio of the curved surface area of the cylinder and the surface area of the sphere?
iii. what is the ratio of the volumes of the cylinder and the sphere? (Textbook pg. no. 141)
Solution:
Radius of base of cylinder = radius of sphere
∴ Radius of sphere = r
Also, height of cylinder = diameter of sphere
∴ h = d
∴ h = 2r …(i)
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.1
∴ radius of sphere : radius of cylinder = 1 : 1.
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.1
∴ curved surface area of cylinder : surface area of sphere = 1:1.
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.1
∴ volume of cylinder : volume of sphere = 3 : 2.

Question 4.
Finding volume of a sphere using cylindrical beaker and water. (Textbook, pg. no. 142)
Maharashtra Board Class 10 Maths Solutions Chapter 7 Mensuration Practice Set 7.1
i. Take a ball and a beaker of the same radius.
ii. Cut a strip of paper of length equal to the diameter of the beaker.
iii. Draw two lines on the strip dividing it into three equal parts.
iv. Stick this strip on the beaker straight up from the bottom.
v. Fill the water in the beaker upto the first mark of the strip from bottom.
vi. Push the ball in the beaker so that it touches the bottom.
Observe how much water level rises.
You will notice that the water level has risen exactly upto the total height of the strip. Try to obtain the formula for volume of sphere using the volume of the cylindrical beaker.
Solution:
Suppose volume of beaker upto height 2r is V.
V = πr2 h
∴ V = πr2(2r) …[∵ h = 2r]
∴ V = 2πr3
But, V = volume of the ball + volume of water in the beaker
∴ 2πr3 = Volume of the ball + \(\frac { 1 }{ 3 } \) × 2πr3
∴ Volume of the ball = 2πr3 – \(\frac { 2 }{ 3 } \) πr3
= \(\frac{6 \pi r^{3}-2 \pi r^{3}}{3}\)
∴ Volume of the ball = \(\frac { 4 }{ 3 } \) πr3
∴ Volume of a sphere = \(\frac { 4 }{ 3 } \) πr3

Maharashtra Board Class 10 Maths Solutions

Maharashtra Board Class 9 Maths Solutions Chapter 3 Triangles Practice Set 3.1

Maharashtra State Board Class 9 Maths Solutions Chapter 3 Triangles Practice Set 3.1

Practice Set 3.1 Geometry 9th Standard Question 1.
In the adjoining figure, ∠ACD is an exterior angle of ∆ABC. ∠B = 40°, ∠A = 70°. Find the measure of ∠ACD.
Maharashtra Board Class 9 Maths Solutions Chapter 3 Triangles Practice Set 3.1 1
Solution:
∠A = 70° , ∠B = 40° [Given]
∠ACD is an exterior angle of ∆ABC. [Given]
∴ ∠ACD = ∠A + ∠B
= 70° + 40°
∴ ∠ACD = 110°

Question 2.
In ∆PQR, ∠P = 70°, ∠Q = 65°, then find ∠R.
Solution:
∠P = 70°, ∠Q = 65° [Given]
In ∆PQR,
∠P + ∠Q + ∠R = 180° [Sum of the measures of the angles of a triangle is 180°]
∴ 70° + 65° + ∠R = 180°
∴ ∠R = 180° – 70° – 65°
∴ ∠R = 45°

Practice Set 3.1 Geometry 9th Question 3.
The measures of angles of a triangle are x°, (x – 20)°, (x – 40)°. Find the measure of each angle.
Solution:
The measures of the angles of a triangle are x°, (x – 20)°, (x – 40)°. [Given]
∴ x°+ (x – 20)° + (x – 40)° = 180° [Sum of the measures of the angles of a triangle is 180°]
∴ 3x – 60 = 180
∴ 3x = 180 + 60
∴ 3x = 240
∴ x = 240
∴ x = \(\frac { 240 }{ 3 }\)
∴ x = 80°
∴ The measures of the remaining angles are
x – 20° = 80° – 20° = 60°,
x – 40° = 80° – 40° = 40°
∴ The measures of the angles of the triangle are 80°, 60° and 40°.

9th Class Geometry Practice Set 3.1 Question 4.
The measure of one of the angles of a triangle is twice the measure of its smallest angle and the measure of the other is thrice the measure of the smallest angle. Find the measures of the three angles.
Solution:
Let the measure of the smallest angle be x°.
One of the angles is twice the measure of the smallest angle.
∴ Measure of that angle = 2x°
Another angle is thrice the measure of the smallest angle.
∴ Measure of that angle = 3x°
∴ The measures of the remaining two angles are 2x° and 3x°.
Now, x° + 2x° + 3x° = 180° [Sum of the measures of the angles of a triangle is 180°]
∴ 6x = 180
∴ x = 180
∴ x = \(\frac { 180 }{ 6 }\)
∴ x° = 30°
The measures of the remaining angles are 2x° = 2 x 30° = 60°
3x° = 3 x 30° = 90°
The measures of the three angles of the triangle are 30°, 60° and 90°.

Question 5.
In the adjoining figure, measures of some angles are given. Using the measures, find the values of x, y, z.
Maharashtra Board Class 9 Maths Solutions Chapter 3 Triangles Practice Set 3.1 2
Solution:
i. ∠NET = 100° and ∠EMR = 140°
∠EMN + ∠EMR = 180°
∴ z +140° =180°
∴ z = 180° -140°
∴ z = 40°

ii. Also, ∠NET + ∠NEM = 180° [Angles in a linear pair]
∴ 100° + y = 180°
∴ y = 180° – 100°
∴ y = 80°

iii. In ∆ENM,
∴ ∠ENM + ∠NEM + ∠EMN = 180° [Sum of the measures of the angles of a triangle is 180°]
∴ x +80°+ 40°= 180°
∴ x = 180° – 80° – 40°
∴ x = 60°
∴ x = 60°, = 80°, z = 40°

Question 6.
In the adjoining figure, line AB || line DE. Find the measures of ∠DRE and ∠ARE using given measures of some angles.
Maharashtra Board Class 9 Maths Solutions Chapter 3 Triangles Practice Set 3.1 3
Solution:
i. ∠B AD = 70°, ∠DER = 40° [Given]
line AB || line DE and seg AD is their transversal.
∴ ∠EDA = ∠BAD [Alternate Angles]
∴ ∠EDA = 70° ….(i)
In ∆DRE,
∠EDR + ∠DER + ∠DRE = 180° [Sum of the measures of the angles of a triangle is 180°]
∴ 70°+ 40° +∠DRE = 180° [From (i) and D – R – A]
∴ ∠DRE = 180° -70° -40°
∴ ∠DRE = 70°

ii. ∠DRE + ∠ARE = 180° [Angles in a linear pair]
∴ 70° + ∠ARE = 180°
∴ ∠ARE = 180°-70°
∴ ∠ARE =110°
∴ ∠DRE = 70°, ∠ARE = 110°

Triangles Class 9 Practice Set 3.1 Question 7.
In ∆ABC, bisectors of ∠A and ∠B intersect at point O. If ∠C = 70°, find the measure of ∠AOB.
Solution:
∠OAB = ∠OAC = – ∠BAC ….(i) [Seg AO bisects ∠BAC]
∠OBA = ∠OBC = – ∠ABC …..(ii) [Seg RO bisects ∠ABC]
In AABC,
∠BAC + ∠ABC + ∠ACB = 180° [Sum of the measures of the angles of a triangle is 180°]
Maharashtra Board Class 9 Maths Solutions Chapter 3 Triangles Practice Set 3.1 4
∴ ∠BAC + ∠ABC + 70° = 180°
∴ ∠BAC + ∠ABC = 180°- 70°
∴ ∠BAC + ∠ABC = 110°
∴ \(\frac { 1 }{ 2 }\)(∠BAC) + \(\frac { 1 }{ 2 }\) (∠ABC) = \(\frac { 1 }{ 2 }\) x 110° [MuItiplying both sides by \(\frac { 1 }{ 2 }\)]
∴ ∠OAB + ∠OBA = 55° ….(iii) [From (i) and (ii)]
In AOAB,
∠OAB + ∠OBA + ∠AOB = 180° [Sum of the measures of the angles of a triangle is 180°]
∴ 55° + ∠AOB = 180° [From (iii)]
∴ ∠AOB = 180°- 55°
∴ ∠AOB = 125°

Question 8.
In the adjoining figure, line AB || line CD and line PQ is the transversal. Ray PT and ray QT are bisectors of ∠BPQ and ∠PQD respectively. Prove that m ∠PTQ = 90°.
Maharashtra Board Class 9 Maths Solutions Chapter 3 Triangles Practice Set 3.1 5
Given: line AB || line CD and line PQ is the transversal.
ray PT and ray QT are the bisectors of ∠BPQ and ∠PQD respectively.
To prove: m∠PTQ = 90°
Solution:
Proof:
∠TPB = ∠TPQ = \(\frac { 1 }{ 2 }\)∠BPQ …(i) [Ray PT bisects ∠BPQ]
∠TQD = ∠TQP = \(\frac { 1 }{ 2 }\)∠PQD ….(ii) [Ray QT bisects ∠PQD]
line AB || line CD and line PQ is their transversal. [Given]
∴∠BPQ + ∠PQD = 180° [Interior angles]
∴ \(\frac { 1 }{ 2 }\) (∠BPQ) + \(\frac { 1 }{ 2 }\) (∠PQD) = \(\frac { 1 }{ 2 }\) x 180° [Multiplying both sides by \(\frac { 1 }{ 2 }\)]
∠TPQ + ∠TQP = 90°
In ∆PTQ,
∠TPQ + ∠TQP + ∠PTQ = 180° [Sum of the measures of the angles of a triangle is 180°]
∴ 90° + ∠PTQ = 180° [From (iii)]
∴ ∠PTQ = 180° – 90°
= 90°
∴ m∠PTQ = 90°

Triangle Practice Set 3.1 Question 9.
Using the information in the adjoining figure, find the measures of ∠a, ∠b and ∠c.
Maharashtra Board Class 9 Maths Solutions Chapter 3 Triangles Practice Set 3.1 6
Solution:
i. ∠c + 100° = 180° [Angles in a linear pair]
∴ ∠c = 180° – 100°
∴ ∠c = 80°

ii. ∠b = 70° [Vertically opposite angles]
iii. ∠a + ∠b +∠c = 180° [Sum of the measures of the angles of a triangle is 180°]
∠a + 70° + 80° = 1800
∴ ∠a = 180° – 70° – 80°
∴ ∠a = 30°
∴ ∠a = 30°, ∠b = 70°,∠ c = 80°

Practice Set 3.1 Geometry Question 10.
In the adjoining figure, line DE || line GF, ray EG and ray FG are bisectors of ∠DEF and ∠DFM respectively. Prove that,
Maharashtra Board Class 9 Maths Solutions Chapter 3 Triangles Practice Set 3.1 7
i. ∠DEG = \(\frac { 1 }{ 2 }\) ∠EDF
ii. EF = FG
Solution:
i. ∠DEG = ∠FEG = x° ….(i) [Ray EG bisects ∠DEF]
∠GFD = ∠GFM = y° …..(ii) [Ray FG bisects ∠DFM]
line DE || line GF and DF is their transversal. [Given]
Maharashtra Board Class 9 Maths Solutions Chapter 3 Triangles Practice Set 3.1 8
∴ ∠EDF = ∠GFD [Alternate angles]
∴ ∠EDF = y° ….(iii) [From (ii)]
line DE || line GF and EM is their transversal. [Given]
Maharashtra Board Class 9 Maths Solutions Chapter 3 Triangles Practice Set 3.1 9
∴ ∠DEF = ∠GFM [Corresponding angles]
∴ ∠DEG + ∠FEG = ∠GFM [Angle addition property]
∴ x°+ x° = y° [From (i) and (ii)]
∴ 2x° = y°
∴ x° = \(\frac { 1 }{ 2 }\)y°
∴ ∠DEG = \(\frac { 1 }{ 2 }\)∠EDF [From (i) and (iii)]

ii. line DE || line GF and GE is their transversal. [Given]
Maharashtra Board Class 9 Maths Solutions Chapter 3 Triangles Practice Set 3.1 11
∴ ∠DEG = ∠FGE …(iv) [Alternate angles]
∴ ∠FEG = ∠FGE ….(v) [From (i) and (iv)]
∴ In ∆FEG,
∠FEG = ∠FGE [From (v)]
∴ EF = FG [Converse of isosceles triangle theorem]

Maharashtra Board Class 9 Maths Chapter 3 Triangles Practice Set 3.1 Intext Questions and Activities

Class 9 Geometry Practice Set 3.1 Question 1. Can you give an alternative proof of the above theorem by drawing a line through point R and parallel to seg PQ in the above figure? (Textbook pg. no. 25)
Maharashtra Board Class 9 Maths Solutions Chapter 3 Triangles Practice Set 3.1 12
Solution:
Yes.
Construction: Draw line RM parallel to seg PQ through a point R.
Proof:
seg PQ || line RM and seg PR is their transversal. [Construction]
∴ ∠PRM = ∠QPR ……..(i) [Alternate angles]
seg PQ || line RM and seg QR is their transversal. [Construction]
∴ ∠SRM = ∠PQR ……..(ii) [Corresponding angles]
∴ ∠PRM + ∠SRM = ∠QPR + ∠PQR [Adding (i) and (ii)]
∴ ∠PRS = ∠PQR + ∠QPR [Angle addition property]

3 Triangles Question 2. Observe the given figure and find the measures of ∠PRS and ∠RTS. (Textbook pg. no.25)
Maharashtra Board Class 9 Maths Solutions Chapter 3 Triangles Practice Set 3.1 13
Solution:
∠PRS is an exterior angle of ∆PQR.
So from the theorem of remote interior angles,
∠PRS = ∠PQR + ∠QPR
= 40° + 30°
∴ ∠PRS = 70°
∴ ∠TRS=70° …[P – T – R]
In ∆RTS,
∠TRS + ∠RTS + ∠TSR = 180° …[Sum of the measures of the angles of a triangle is 180°]
∴ 70° + ∠RTS + 20° = 180°
∴ ∠RTS + 90° = 180°
∴ ∠RTS = 180°
∴ ∠RTS = 90°

9th Class Geometry Triangles Question 3. In the given figure, bisectors of ∠B and ∠C of ∆ABC intersect at point P. Prove that ∠BPC = 90° + \(\frac { 1 }{ 2 }\)∠BAC.
Complete the proof by filling in the blanks. (Textbook pg. no.27)
Maharashtra Board Class 9 Maths Solutions Chapter 3 Triangles Practice Set 3.1 14
Solution:
Proof:
In ∆ABC,
∠BAC + ∠ABC + ∠ACB = 180° …[Sum of the measures of the angles of a triangle is 180°]
∴ ∠BAC + – ∠ABC + ∠ACB = 180 … [Multiplying each term by \(\frac { 1 }{ 2 }\)]
∴ ∠BAC + ∠PBC + ∠PCB = 90°
∴ ∠PBC + ∠PCB = 90° – 1 ∠BAC ………(i)
In∆BPC,
∠BPC + ∠PBC + ∠PCB = 180° …….[Sum of measures of angles of a triangle]
∴ ∠BPC + 90° – \(\frac { 1 }{ 2 }\)∠BAC = 180° ……[From (i)]
∴ ∠BPC = 180° – 90°\(\frac { 1 }{ 2 }\)∠BAC
= 180°- 90°+ \(\frac { 1 }{ 2 }\)∠BAC
= 90°+ \(\frac { 1 }{ 2 }\)∠BAC

Maharashtra Board Class 9 Maths Solutions

Maharashtra Board Class 6 Maths Solutions Chapter 11 Ratio-Proportion Practice Set 28

Maharashtra State Board Class 6 Maths Solutions Chapter 11 Ratio-Proportion Practice Set 28

6th Standard Maths Practice Set 28 Question 1.
In each example below, find the ratio of the first number to the second:
i. 24, 56
ii. 63,49
iii. 52, 65
iv. 84, 60
v. 35, 65
vi. 121, 99
Solution:
i. 24, 56
\(\frac{24}{56}=\frac{24 \div 8}{56 \div 8}=\frac{3}{7}\)
= 3:7

ii. 63,49
\(\frac{63}{49}=\frac{63 \div 7}{49 \div 7}=\frac{9}{7}\)
= 9:7

iii. 52, 65
\(\frac{52}{65}=\frac{52 \div 13}{65 \div 13}=\frac{4}{5}\)
= 4:5

iv. 84, 60
\(\frac{84}{60}=\frac{84 \div 12}{60 \div 12}=\frac{7}{5}\)
= 7:5

v. 35, 65
\(\frac{35}{65}=\frac{35 \div 5}{65 \div 5}=\frac{7}{13}\)
= 7:13

vi. 121, 99
\(\frac{121}{99}=\frac{121 \div 11}{99 \div 11}=\frac{11}{9}\)
= 11:9

6th Maths Practice Set 28 Question 2.
Find the ratio of the first quantity to the second.
i. 25 beads, 40 beads
ii. Rs 40, Rs 120
iii. 15 minutes, 1 hour
iv. 30 litres, 24 litres
v. 99 kg, 44000 grams
vi. 1 litre, 250 ml
vii. 60 paise, 1 rupee
viii. 750 grams, \(\frac { 1 }{ 2 }\) kg
ix. 125 cm, 1 metre
Solution:
i. Required Ratio = \(\frac{25}{40}=\frac{25 \div 5}{40 \div 5}=\frac{5}{8}\)

ii. Required Ratio = \(\frac{40}{120}=\frac{40 \div 40}{120 \div 40}=\frac{1}{3}\)

iii. 1 hour = 60 minutes
Required Ratio = \(\frac{15}{60}=\frac{15 \div 15}{60 \div 15}=\frac{1}{4}\)

iv. Required Ratio = \(\frac{30}{24}=\frac{30 \div 6}{24 \div 6}=\frac{5}{4}\)

v. 99 kg = 99 x 1000 grams = 99000 grams
Required Ratio = \(\frac{99000}{44000}=\frac{99000 \div 1000}{44000 \div 1000}=\frac{99}{44}\)
= \(\frac{99}{44}=\frac{99 \div 11}{44 \div 11}=\frac{9}{4}\)

vi. 1 litre, 250 ml
1 litre = 1000 ml
Required Ratio = \(\frac{1000}{250}=\frac{1000 \div 10}{250 \div 10}=\frac{100}{25}\)
= \(\frac{100}{25}=\frac{100 \div 25}{25 \div 25}=\frac{4}{1}\)

viii. 750 grams, \(\frac { 1 }{ 2 }\) kg
\(\frac { 1 }{ 2 }\) kg = \(\frac { 1000 }{ 2 }\) grams = 500 grams
Required Ratio = \(\frac{750}{500}=\frac{750 \div 10}{500 \div 10}=\frac{75}{50}\)
= \(\frac{75}{50}=\frac{75 \div 25}{50 \div 25}=\frac{3}{2}\)

ix. 125 cm, 1 metre
1 metre = 100 cm
Required Ratio = \(\frac{125}{100}=\frac{125 \div 25}{100 \div 25}=\frac{5}{4}\)

6th Std Maths Practice Set 28 Question 3.
Reema has 24 notebooks and 18 books. Find the ratio of notebooks to books.
Solution:
Ratio of notebooks to books
Maharashtra Board Class 6 Maths Solutions Chapter 11 Ratio-Proportion Practice Set 28 1
∴ The ratio of notebooks to books with Reema is \(\frac { 4 }{ 3 }\)

Practice Set 28 Question 4.
30 cricket players and 20 kho-kho players are training on a field. What is the ratio of cricket players to the total number of players?
Solution:
Total number of players = Cricket players + Kho-kho players
= 30 + 20 = 50
Ratio of cricket players to the total number of players
Maharashtra Board Class 6 Maths Solutions Chapter 11 Ratio-Proportion Practice Set 28 2
∴ The ratio of cricket players to the total number of players is \(\frac { 3 }{ 5 }\).

Question 5.
Snehal has a red ribbon that is 80 cm long and a blue ribbon 2.20 m long. What is the ratio of the length of the red ribbon to that of the blue ribbon?
Solution:
1 metre =100 cm
Length of the red ribbon = 80 cm
Length of the blue ribbon = 2.20 m = 2.20 x 100 cm
\(=\frac{220}{100} \times \frac{100}{1}=\frac{220 \times 100}{100 \times 1}\)
= 220 cm
∴ Ratio of length of the red ribbon to that of the blue ribbon
Maharashtra Board Class 6 Maths Solutions Chapter 11 Ratio-Proportion Practice Set 28 3
∴ The ratio of the length of the red ribbon to that of the blue ribbon is \(\frac { 4 }{ 11 }\).

11 Ratio Question 6.
Shubham’s age today is 12 years and his father’s is 42 years. Shubham’s mother is younger than his father by 6 years. Find the following ratios.
i. Ratio of Shubham’s age today to his mother’s age today.
ii. Ratio of Shubham’s mother’s age today to his father’s age today.
iii. The ratio of Shubham’s age to his mother’s age when Shubham was 10 years old.
Solution:
Shubham’s age today = 12 years
Shubham’s father’s age = 42 years
Shubham’s mother age = Shubham’s father’s age – 6 years
= 42 years – 6 years = 36 years

i. Ratio of Shubham’s age today to his mother’s age today
Maharashtra Board Class 6 Maths Solutions Chapter 11 Ratio-Proportion Practice Set 28 4
∴ The ratio of Shubham’s age today to his mother’s age today is \(\frac { 1 }{ 3 }\).

ii. Ratio of Shubham’s mother age today to his father’s age today
Maharashtra Board Class 6 Maths Solutions Chapter 11 Ratio-Proportion Practice Set 28 5
∴ The ratio of Shubham’s mother’s age today to his father’s age today is \(\frac { 6 }{ 7 }\).

iii. Shubham’s age today is 12 years and his mothers age is 36 years.
Hence when Shubham’s age was 10 years, his mother’s age was 34 years (i.e. 36 – 2 years).
Ratio of Shubham’s age to his mother’s age when Shubham was 10 years old
Maharashtra Board Class 6 Maths Solutions Chapter 11 Ratio-Proportion Practice Set 28 6
∴ The ratio of Shubham’s age to his mother’s age when Shubham was 10 years old is \(\frac { 5 }{ 17 }\)

Maharashtra Board Class 6 Maths Chapter 11 Ratio-Proportion Practice Set 28 Intext Questions and Activities

Question 1.
In the figure, colour some squares with any colour you like and leave some blank. (Textbook pg. no. 57)
Maharashtra Board Class 6 Maths Solutions Chapter 11 Ratio-Proportion Practice Set 28 7
i. Count all the boxes and write the number.
ii. Count the colored ones and write the number.
iii. Count the blank ones and write the number.
iv. Find the ratio of the colored boxes to the blank ones.
v. Find the ratio of the colored boxes to the total boxes.
vi. Find the ratio of the blank boxes to the total boxes.
Solution:
i. The number of all boxes is 16.
ii. The number of colored boxes is 7.
iii. The number of blank boxes is 9.
iv. Ratio of the colored boxes to the blank ones
Maharashtra Board Class 6 Maths Solutions Chapter 11 Ratio-Proportion Practice Set 28 8
v. Ratio of the colored boxes to the total boxes
Maharashtra Board Class 6 Maths Solutions Chapter 11 Ratio-Proportion Practice Set 28 9
vi. Ratio of the blank boxes to the total boxes
Maharashtra Board Class 6 Maths Solutions Chapter 11 Ratio-Proportion Practice Set 28 10

Maharashtra Board Class 6 Maths Solutions

Maharashtra Board Class 7 Maths Solutions Chapter 10 Banks and Simple Interest Practice Set 40

Maharashtra State Board Class 7 Maths Solutions Chapter 10 Banks and Simple Interest Practice Set 40

Question 1.
If Rihanna deposits Rs 1500 in the school fund at 9 p.c.p.a for 2 years, what is the total amount she will get?
Solution:
Here, P = Rs 1500, R = 9 p.c.p.a , T = 2 years
∴ Total interest = \(\frac{\mathrm{P} \times \mathrm{R} \times \mathrm{T}}{100}\)
= \(\frac{1500 \times 9 \times 2}{100}\)
= 1500 x 9 x 2
= Rs 270
∴ Total amount = Principal + Interest
= 1500 + 270
= Rs 1770
∴ Rihanna will get a total amount of Rs 1770.

Question 2.
Jethalal took a housing loan of Rs 2,50,000 from a bank at 10 p.c.p.a. for 5 years. What is the yearly interest he must pay and the total amount he returns to the bank?
Solution:
Here, P = Rs 250000, R = 10 p.c.p.a., T = 5 years
∴ Total interest = \(\frac{\mathrm{P} \times \mathrm{R} \times \mathrm{T}}{100}\)
= \(\frac{250000 \times 10 \times 5}{100}\)
= 2500 x 10 x 5
= Rs 1,25,000
∴ Yearly interest = Total interest ÷ Time = 1,25,000 ÷ 5 = Rs 25000
Total amount to be returned = Principal + Total interest
= 250000 + 125000 = Rs 375000
∴ The yearly interest is Rs 25,000 and Jethalal will have to return Rs 3,75,000 to the bank.

Question 3.
Shrikant deposited Rs 85,000 for \(2\frac { 1 }{ 2 }\) years at 7 p.c.p.a. in a savings bank account. What is the total
interest he received at the end of the period?
Solution:
Here, P = Rs 85000, R = 7 p.c.p.a., T = \(2\frac { 1 }{ 2 }\) years = 2.5 years
∴ Total interest = \(\frac{\mathrm{P} \times \mathrm{R} \times \mathrm{T}}{100}\)
= \(\frac{85000 \times 7 \times 2.5}{100}\)
= \(\frac{85000 \times 7 \times 25}{100 \times 10}\)
= 85 x 7 x 25
= Rs 14875
∴ The total interest received by Shrikant at the end of the period is Rs 14875.

Question 4.
At a certain rate of interest, the interest after 4 years on Rs 5000 principal is Rs 1200. What would be the interest on Rs 15000 at the same rate of interest for the same period?
Solution:
The interest on Rs 5000 after 4 years is Rs 1200.
Let us suppose the interest on Rs 15000 at the same rate after 4 years is Rs x.
Taking the ratio of interest and principal, we get
∴ \(\frac{x}{15000}=\frac{1200}{5000}\)
∴ \(x=\frac{1200 \times 15000}{5000}\)
= Rs 3600
∴ The interest received on Rs 15000 is Rs 3600.

Question 5.
If Pankaj deposits Rs 1,50,000 in a bank at 10 p.c.p.a. for two years, what is the total amount he will get from the bank?
Solution:
Here, P = 150000, R = 10 p.c.p.a., T = 2 years
∴ Total interest = \(\frac{\mathrm{P} \times \mathrm{R} \times \mathrm{T}}{100}\)
= \(\frac{150000 \times 10 \times 2}{100}\)
= Rs 30000
∴ Total amount = Principal + Total Interest
= 150000 + 30000
= Rs 180000
∴ Pankaj will receive Rs 180000 from the bank.

Maharashtra Board Class 7 Maths Chapter 10 Banks and Simple Interest Practice Set 40 Intext Questions and Activities

Question 1.
Observe the entries made in the page of a passbook shown below and answer the following questions. (Textbook pg. no. 70)

Maharashtra Board Class 7 Maths Solutions Chapter 10 Banks and Simple Interest Practice Set 40 1

  1. On 2.2.16 the amount deposited was Rs__and the balance Rs__.
  2. On 12.2.16, Rs__were withdrawn by cheque no. 243965. The balance was Rs__
  3. On 26.2.2016 the bank paid an interest of Rs__

Solution:

  1. 1500, 7000
  2. 3000, 9000
  3. 135

Practice Set 40 Class 7 Question 2.
Suvidya borrowed a sum of Rs 30000 at 8 p.c.p.a. interest for a year from her bank to buy a computer. At the end of the period, she had to pay back an amount of Rs 2400 over and above what she had borrowed.
Based on this information fill in the blanks below. (Textbook pg. no. 70)

  1. Principal = Rs__
  2. Rate of interest =__%
  3. Interest = Rs__
  4. Time =__year.
  5. The total amount returned to the bank = 30,000 + 2,400 = Rs__

Solution:

  1. 30000
  2. 8
  3. 2400
  4. 1
  5. Rs 32400

Maharashtra Board Class 7 Maths Solutions

Maharashtra Board Class 9 Maths Solutions Chapter 4 Ratio and Proportion Practice Set 4.5

Maharashtra State Board Class 9 Maths Solutions Chapter 4 Ratio and Proportion Practice Set 4.5

Question 1.
Which number should be subtracted from 12, 16 and 21 so that resultant numbers are in continued proportion?
Solution:
Let the number to be subtracted be x.
∴ (12 – x), (16 – x) and (21 – x) are in continued proportion.
Maharashtra Board Class 9 Maths Solutions Chapter 4 Ratio and Proportion Practice Set 4.5 1
∴ 84 – 4x = 80 – 5x
∴ 5x – 4x = 80 – 84
∴ x = -4
∴ -4 should be subtracted from 12,16 and 21 so that the resultant numbers in continued proportion.

Question 2.
If (28 – x) is the mean proportional of (23 – x) and (19 – x), then find the value ofx.
Solution:
(28 – x) is the mean proportional of (23 – x) and (19-x). …[Given]
Maharashtra Board Class 9 Maths Solutions Chapter 4 Ratio and Proportion Practice Set 4.5 2
∴ -5(19 – x) = 9(28 – x)
∴ -95 + 5x = 252 – 9x
∴ 5x + 9x = 252 + 95
∴ 14x = 347
∴ x = \(\frac { 347 }{ 14 }\)

Question 3.
Three numbers are in continued proportion, whose mean proportional is 12 and the sum of the remaining two numbers is 26, then find these numbers.
Solution:
Let the first number be x.
∴ Third number = 26 – x
12 is the mean proportional of x and (26 – x).
∴ \(\frac { x }{ 12 }\) = \(\frac { 12 }{ 26 – x }\)
∴ x(26 – x) = 12 x 12
∴ 26x – x2 = 144
∴ x2 – 26x + 144 = 0
∴ x2 – 18x – 8x + 144 = 0
∴ x(x – 18) – 8(x – 18) = 0
∴ (x – 18) (x – 8) = 0
∴ x = 18 or x = 8
∴ Third number = 26 – x = 26 – 18 = 8 or 26 – x = 26 – 8 = 18
∴ The numbers are 18, 12, 8 or 8, 12, 18.

Question 4.
If (a + b + c)(a – b + c) = a2 + b2 + c2, show that a, b, c are in continued proportion.
Solution:
(a + b + c)(a – b + c) = a2 + b2 + c2 …[Given]
∴ a(a – b + c) + b(a – b + c) + c(a – b + c) = a2 + b2 + c2
∴ a2 – ab + ac + ab – b2 + be + ac – be + c2 = a2 + b2 + c2
∴ a2 + 2ac – b2 + c2 = a2 + b2 + c2
∴ 2ac – b2 = b2
∴ 2ac = 2b2
∴ ac = b2
∴ b2 = ac
∴ a, b, c are in continued proportion.

Question 5.
If \(\frac { a }{ b }\) = \(\frac { b }{ c }\) and a, b, c > 0, then show that,
i. (a + b + c)(b – c) = ab – c2
ii. (a2 + b2)(b2 + c2) = (ab + be)2
iii. \(\frac{a^{2}+b^{2}}{a b}=\frac{a+c}{b}\)
Solution:
Let \(\frac { a }{ b }\) = \(\frac { b }{ c }\) = k
∴ b = ck
∴ a = bk =(ck)k
∴ a = ck2 …(ii)

i. (a + b + c)(b – c) = ab – c2
L.H.S = (a + b + c) (b – c)
= [ck2 + ck + c] [ck – c] … [From (i) and (ii)]
= c(k2 + k + 1) c (k – 1)
= c2 (k2 + k + 1) (k – 1)
R.H.S = ab – c2
= (ck2) (ck) – c2 … [From (i) and (ii)]
= c2k3 – c2
= c2(k3 – 1)
= c2 (k – 1) (k2 + k + 1) … [a3 – b3 = (a – b) (a2 + ab + b2]
∴ L.H.S = R.H.S
∴ (a + b + c) (b – c) = ab – c2

ii. (a2 + b2)(b2 + c2) = (ab + bc)2
b = ck; a = ck2
L.H.S = (a2 + b2) (b2 + c2)
= [(ck2) + (ck)2] [(ck)2 + c2] … [From (i) and (ii)]
= [c2k4 + c2k2] [c2k2 + c2]
= c2k2 (k2 + 1) c2 (k2 + 1)
= c4k2 (k2 + 1)2
R.H.S = (ab + bc)2
= [(ck2) (ck) + (ck)c]2 …[From (i) and (ii)]
= [c2k3 + c2k]2
= [c2k (k2 + 1)]2 = c4(k2 + 1)2
∴ L.H.S = R.H.S
∴ (a2 + b2) (b2 + c2) = (ab + bc)2

iii. \(\frac{a^{2}+b^{2}}{a b}=\frac{a+c}{b}\)
Maharashtra Board Class 9 Maths Solutions Chapter 4 Ratio and Proportion Practice Set 4.5 3

9th Standard Algebra Practice Set 4.5 Question 6. Find mean proportional of \(\frac{x+y}{x-y}, \frac{x^{2}-y^{2}}{x^{2} y^{2}}\).
Solution:
Let a be the mean proportional of \(\frac{x+y}{x-y}\) and \(\frac{x^{2}-y^{2}}{x^{2} y^{2}}\)
Maharashtra Board Class 9 Maths Solutions Chapter 4 Ratio and Proportion Practice Set 4.5 4

Maharashtra Board Class 9 Maths Solutions

Maharashtra Board Class 8 Maths Solutions Chapter 9 Discount and Commission Practice Set 9.1

Maharashtra State Board Class 8 Maths Solutions Chapter 9 Discount and Commission Practice Set 9.1

8th Standard Maths Practice Set 9.1 Question 1. If marked price = Rs 1700, selling price = Rs 1540, then find the discount.
Solution:
Here, Marked price = Rs 1700,
selling price = Rs 1540
Selling price = Marked price – Discount
∴ 1540 = 1700 – Discount
∴ Discount = 1700 – 1540
= Rs 160
∴ The amount of discount is Rs 160.

Discount and Commission Practice Set 9.1 Question 2. If marked price Rs 990 and percentage of discount is 10, then find the selling price.
Solution:
Here, marked price = Rs 990,
discount = 10%
Let the percentage of discount be x
∴ x = 10%
i. Discount
Maharashtra Board Class 8 Maths Solutions Chapter 9 Discount and Commission Practice Set 9.1 1
= Rs 99

ii. Selling price = Marked price – Discount
= 990 – 99
= Rs 891
∴ The selling price is Rs 891.

Practice Set 9.1 Question 3. If selling price Rs 900, discount is 20%, then find the marked price.
Solution:
Here, selling price = Rs 900, discount = 20%
Let the marked price be Rs 100
Since, the discount given = 20%
∴ Amount of discount = Rs 20
∴ Selling price = 100 – 20 – Rs 80
Let actual marked price be Rs x
∴ For marked price of Rs x, selling price is Rs 900
\(\frac{80}{100}=\frac{900}{x}\)
∴ 80 × x = 100 × 900
∴ \(x=\frac{100 \times 900}{80}\)
∴ x = Rs 1125
∴ The marked price is Rs 1125.

Discount and Commission Std 8 Question 4. The marked price of the fan is Rs 3000. Shopkeeper gave 12% discount on it. Find the total discount and selling price of the fan.
Solution:
Here, Marked price = Rs 3000, discount = 12%
Let the percentage of discount be x.
∴ x = 12%
i. Discount
Maharashtra Board Class 8 Maths Solutions Chapter 9 Discount and Commission Practice Set 9.1 2
= 30 × 12
= Rs 360

ii. Selling price = Marked price – Discount
= 3000 – 360
= Rs 2640
∴ The total discount is Rs 360 and the selling price of the fan is Rs 2640.

Discount and Commission 8th Standard Question 5. The marked price of a mixer is Rs 2300. A customer purchased it for Rs 1955. Find percentage of discount offered to the customer.
Solution:
Here, marked price = Rs 2300,
selling price = Rs 1955
i. Selling price = Marked price – Discount
∴ 1955 = 2300 – Discount
∴ Discount = 2300 – 1955
= Rs 345

ii. Let the percentage of discount be x
Maharashtra Board Class 8 Maths Solutions Chapter 9 Discount and Commission Practice Set 9.1 3
Maharashtra Board Class 8 Maths Solutions Chapter 9 Discount and Commission Practice Set 9.1 4
∴ x = 15%
∴ The percentage of discount offered to the customer is 15%.

Question 6.
A shopkeeper gives 11% discount on a television set, hence the cost price of it is Rs 22,250. Then find the marked price of the television set.
Solution:
Here, selling price = Rs 22,250, discount = 11%
Let marked price be Rs 100
Since, the discount given = 11%
∴ Amount of discount = Rs 11
∴ Selling price = 100 – 11 = Rs 89
∴ Let actual marked price be Rs x
∴ For marked price of Rs x, selling price is Rs 22,250
Maharashtra Board Class 8 Maths Solutions Chapter 9 Discount and Commission Practice Set 9.1 5
∴ x = Rs 25,000
∴ The marked price of the television set is Rs 25,000.

8th Std Maths Discount and Commission Question 7. After offering discount of 10% on marked price, a customer gets total discount of Rs 17. To find the cost price for the customer, fill in the following boxes with appropriate numbers and complete the activity.
Solution:
Suppose, marked price of the item = 100 rupees Therefore, for customer that item costs 100 – 10 = 90 rupees.
Hence, when the discount is [10] then the selling price is [90] rupees.
Suppose when the discount is [17] rupees, the selling price is x rupees.
Maharashtra Board Class 8 Maths Solutions Chapter 9 Discount and Commission Practice Set 9.1 6
∴ The customer will get the item for Rs 153.

Question 8.
A shopkeeper decides to sell a certain item at a certain price. He tags the price on the item by increasing the decided price by 25%. While selling the item, he offers 20% discount. Find how many more or less percent he gets on the decided price.
Solution:
Here, price increase = 25%,
discount offered = 20%
Let the decided price be Rs 100
∴ Increase in price = Rs 25
∴ Shopkeeper marks the price = 100 + 25
= Rs 125
∴ marked price = Rs 125
Let the percentage of discount be x
∴ x = 20%
Maharashtra Board Class 8 Maths Solutions Chapter 9 Discount and Commission Practice Set 9.1 7
∴ Selling price = Marked price – Discount
= 125 – 25
= Rs 100
∴ If the decided price is Rs 100, then shopkeeper gets Rs 100.
∴ The shopkeeper gets neither more nor less than the decided price i.e. he gets 0% more / less.

Maharashtra Board Class 8 Maths Chapter 9 Discount and Commission Practice Set 9.1 Intext Questions and Activities

Question 1.
Write the appropriate numbers in the following boxes. (Textbook pg. no. 51)

  1. \(\frac { 12 }{ 100 }=\) __ percent = __%
  2. 47% = __
  3. 86% = __
  4. 4% of 300 = 300 × __ = __
  5. 15% of 1700 = 1700 × __= __

Solution:

  1. \(\frac { 12 }{ 100 }=\) 12 percent = 12%
  2. 47% = \(\frac { 47 }{ 100 }\)
  3. 86% = \(\frac { 86 }{ 100 }\)
  4. 4% of 300 = 300 × \(\frac { 4 }{ 100 }\) = 12
  5. 15% of 1700 = 1700 × \(\frac { 15 }{ 100 }\) = 255

Question 2.
You may have seen advertisements like ‘Monsoon Sale’, ‘Stock Clearance Sale’ etc offering different discount. In such a sale, a discount is offered on various goods. Generally in the month of July, sales of clothes are declared. Find and discuss the purpose of such sales. (Textbook pg. no. 51)
Solution:
(Students should attempt the above activity on their own)

Maharashtra Board Class 8 Maths Solutions