By using Ganita Prakash Class 7 Solutions and Part 1 Chapter 4 Expressions using Letter Numbers Class 7 Question Answer, students can improve their problem-solving skills.
Class 7 Maths Chapter 4 Expressions using Letter Numbers Solutions
Ganita Prakash Class 7 Chapter 4 Solutions
Class 7 Maths Ganita Prakash Chapter 4 Solutions Expressions using Letter Numbers
4.1. The Notion of Letter – Numbers
Consider a situation that Shabnam is 3 years older than Aftab.
Page: 81
Question 1.
Consider a situation that Shabnam is 3 years older than Aftab.
Can we write this as an expression?
Answer:
Let Aftab’s age be given to be x years.
Then shabnam’s age = (x + 3) years.
Page: 82
Question 2.
Given the age of Shabnam, write an expression to find Aftab’s age.
Answer:
Let Shabnam’s age be x years.
Since, Shabnam is 3 years older than Aftab.
Therefore, Aftab’s age = (x – 3) years.
Question 3.
Use this expression to find Aftab’s age if Shabnam’s age is 20.
Answer:
Here, Shabnam’s age is 20
Then, Aftab’s age = x – 3
= 20 – 3 (∴, x = 20)
= 17 years.
Consider the situation:
The price of a coconut is ₹35 and the price of 1 kg jaggery is ₹60.
Page: 83
Question 4.
How much should Ketaki pay if she buys 10 coconuts and 5 kg jaggery?
Answer:
Cost of 10 coconuts = 10 × ₹ 35 = ₹ 350
Cost of 5 kg jaggery = 5 × ₹ 60 = ₹ 300
Hence, Ketaki should pay = ₹ 350 + ₹ 300
= ₹ 650.
Question 5.
How much should she pay if she buys 8 coconuts and 9 kg jaggery?
Answer:
‘Cost of 8 coconuts = 8 × ₹ 35 = ₹ 280
Cost of 9 kg jaggery = 9 × ₹ 60 = ₹ 540
Total cost she should pay = ₹ 280 + ₹ 540
= ₹820
Question 6.
Write an algebraic expression to find the total amount to be paid for a given number of coconuts and quantity of jaggery.
Answer:
Let ‘ c ‘ be the number of coconuts purchased and let ‘ j ‘ kg of jaggery be purchased.
∴ , An algebraic expression to find the total amount to be paid for ‘ c ‘ number of the coconuts and ‘ j ‘ kg of jaggery is:
“35 × c + 60 × \(\vec{j}\) ”
Question 7.
Use this expression (or formula) to find the total amount to be paid for 7 coconuts and 4 kg of jaggery.
Answer:
Putting c = 7 and j = 4 in the expression
35 × c + 60 × j = 35 × 7 + 60 × 4
= 245 + 240 = 485
So, the total amount to be paid = ₹485.
Question 8.
What is the perimeter of a square with side length 7 cm? Use the expression to find out.
Answer:
The expression for the perimeter of a square whose length of side is given to be equal to ‘q’ is 4 × q.
Here,
q = 7 cm
So, perimeter = 4 × 7 = 28 cm
Page: 84
Question 9.
Figure it Out:
1. Write formulas for the perimeter of:
(a) triangle with all sides equal.
(b) a regular pentagon (as we have learnt last year, we use the word ‘regular’ to say that all side lengths and angle measures are equal)
(c) a regular hexagon
Answer:
(a) Let the length of one side of triangle be ‘ t ‘.
So, formula for the perimeter = 3 × t
(b) Let the length of one side of a regular pentagon be ‘ p ‘.
So, formula for the perimeter = 5 × p.
(c) Let the length of one side of a regular hexagon be ‘ h ‘.
So, formula for the perimeter = 6 × h.
2. Munirathna has a 20 m long pipe. However, he wants a longer watering pipe for his garden. He joins another pipe of some length to this one. Give the expression for the combined length of the pipe. Use the letter-number ‘ k ‘ to denote the length in meters of the other pipe.
Answer:
Let the length in meters of the other pipe be ‘ k ‘.
Since, it has to be joined with a 20 m long pipe.
So, the combined length of the pipe is given by ” K + 20 “.
3. What is the total amount Krithika has, if she has the following numbers of notes of ₹100, ₹20 and ₹5? Complete the following table:
Answer:
4. Venkatalakshmi owns a flour mill. It takes 10 seconds for the roller mill to start running. Once it is running, each kg of grain takes 8 seconds to grind into powder. Which of the expressions below
describes the time taken to complete grind ‘ y ‘ kg of grain, assuming the machine is off initially?
(a) 10 + 8 + y
(b) (10 + 8) × y
(c) 10 × 8 × y
(d) 10 + 8 × y
(e) 10 × y + 8
Answer:
In the given expressions, (d) 10 + 8 xy, describes the time taken to complete grind ‘ y ‘ kg of grain.
Page: 84, 85
5. Write algebraic expressions using letters of your choice.
(a) 5 more than a number
(b) 4 less than a number
(c) 2 less than 13 times a number
(d) 13 less than 2 times a number
Answer:
(a) Let the number be ‘ n ‘.
So, 5 more than a number is represented by ” n + 5 “.
(b) Let the number be ‘ p ‘
So, 4 less than a number is represented by ” p – 4 “.
(c) Let the number be ‘ x ‘.
So, 2 less than 13 times a number is represented by ” 13 × x – 2 ”
(d) Let the number be ‘ y ‘.
So, 13 less than 2 times a number is represented by ” 2 y – 13 “.
Note: Sometimes we write 2 z for 2 × z in an algebraic expression.
6. Describe situations corresponding to the following algebraic expressions:
(a) 8 × x + 3 × y
(b) 15 × j – 2 × k
Answer:
(a) If price of one pen be ₹8 and that of one pencil be 23, then the algebraic expression for the total cost of ‘ n ‘ pens and ‘ y ‘ pencils.
(b) A shopkeeper earns a profit of ‘ ₹ j ‘ for 15days and looses ‘₹ k’ for 2days, the algebraic expression of for net profit in 17days.
7. In a calendar month, if any 2 × 3 grid full of dates is chosen as shown in the picture, write expressions for the dates in the blank cells if the bottom middle cell has date’ w ‘.
Answer:
immmmmmmmmmmmmmmmm
Look at this number sequence :
4, 8, 12, 16, 20, 24, 28,
Page 86
8. Find an algebraic expression to get the nth term of this sequence.
Answer:
Here, given sequence is :
4,8,12,16,20,24,28, …
We can notice that each term is 4 times its number of position.
So, expression for it’s ‘nth‘ term is:
“4 × n”.
Page 92
Question 10.
Find an algebraic expression to get the nth term of this sequence.
Answer:
Since, values of 5 u and 5 + u are different. Hence, the two expressions are different.
Question 11.
Let us compare the values that these expressions take for different values of y.
After filling in the two diagrams, do you think the two expressions are equal?
Answer:
The values of two expressions for a given value of y are not same. Therefore, the two expressions are not equal.
Page 93, 94
Question 12.
Figure it Out :
1. Add the numbers in each picture below. Write their corresponding expressions and simplify them. Try adding the numbers in each picture in a couple of different ways and see that you get the same thing.
Answer:
Expression 1:
5y + (-6) + x + 2 + 5y + x
= 10y + 2x – 6 + 2
= 10y + 2x – 4
Expression 2:
5y + (-6) + x + 2 + 5y + x
= 10y + 2x – 4
All these expressions are same.
Here, all the expressions at I, II, III and IV.
Expression 1:
(- 5 g) + (5 k) + (5 k) + (- 5 g) + (5 k) + (5 k) + (5 k) + (5 k) + (5 k) + (5 k) + (5 k) + (5 k) + (- 5 g) + (5 k) + (5 k) + (- 5 g)
= (- 5 g) + (- 5 g) + (- 5 g) + (- 5 g) + 5 k + 5 k + 5 k + 5 k + 5 k + 5 k + 5 k + 5 k + 5 k + 5 k + 5 k + 5 k.
= – 20 g + 60 k.
Expression 2:
5 k + 5 k + 5 k + 5 k + 5 k + 5 k + 5 k + 5 k + 5 k + 5 k + 5 k + 5 k + (- 5 g) + (- 5 g) + (- 5 g) + (- 5 g) .
= 60 k – 20 g
Here, expression 1 and expression 2 are same.
2. Simplify each of the following expressions:
(a) p + p + p + p, p + p + p + q, p + q + p – q
(b) p – q + p – q, p + q – p + q
(c) p + q – (p + q), p – q – p – q
(d) 2d – d – d – d, 2d – d – d – c
(e) 2d – d – (d – c), 2d – (d – d) – c
(f) 2d – d – c – c
Answer:
(a) p + p + p + p = 4 × p = 4 p
p + p + p + q = 3 × p + q = 3 p + q
p + q + p – q = 2 × p + q – q = 2 p
(b) p – q + p – q = 2 × p – 2 × q = 2 p – 2 q
p + q – p + q = p – p + 2 × q = 2 q
(c) p + q – (p + q) = p + q – p – q = 0
p + q – p + q = p – p – 2 q = – 2 q
(d) 2d – d – d – d = 2d – 3d = – d
2d – d – d – c = 2d – 2d – c = – c
(e) 2d – d – (d – c) = 2d – d – d + c = c
2d – (d – d) – c = 2d – 0 – c = 2d – c
(f) 2d – d – c – c =d – 2 c
Page 95
Question 14.
Find out the formula of this number machine.
Answer:
∴ , formula is :
Page 96
Question 15.
Find the formulas of the number machines below and write the expression for each set of inputs.
Answer:
Question 16.
Now, make a formula on your own. Write a few number machines as examples using that formula. Challenge your classmates to figure it out!
Answer:
Answer:
a × a + b × b
Page: 99
Question 17.
Consider a set of numbers from the calendar (having endless rows) forming under the following shape:
Find the sum of all the numbers. Compare it with the number in the centre: 15. Repeat this for another set of numbers that forms this shape. Whatdo you observe?
We see that the total sum is always 5 times the number in the centre.
Answer:
Here, the sum of all the numbers
= 8 + 14 + 15 + 16 + 22 = 75
Central number = 15
So, the total sum is 5 times the central number.
Let us consider another set of numbers that forms this shape.
Here, the sum of all the numbers
= 15 + 21 + 22 + 23 + 29 = 110
Central number = 22
Clearly, 110 = 22 × 5
So, here again the total sum obtained is 5 times the central number.
Question 18.
Will this always happen? How do you show this?
Answer:
Yes. This will always happen. This can be explained by forming a similar shape but this time using Letter – Numbers.
If we add all the numbers in the above shape, then we obtain the sum
(a – 7) + (a – 1) + (a) + (a + 1) + (a + 7)
= a + a + a + a + a – 7 – 1 + 1 + 7
= 5 a – 8 + 8
= 5 a
which is 5 times of the central number ‘ a ‘.
Question 19.
Find other shapes for which the sum of the numbers within the figure is always a multiple of one of the numbers.
Answer:
110 = 2 × 22
(a-12) + (a-1) + (a) + (a+1) + (a+12)
= 5 a
Note: Any table made on the basis of the above pattern will have sum of numbers which will always be a multiple of the number within it.
Page: 102, 103, 104
Question 20.
Figure it Out :
1. One plate of Jowar roti costs ₹30 and one plate of Pulao costs ₹20. If x plates of Jowar roti and y plates of Pulao were ordered in a day, which expression(s)describe the total amount in rupees earned that day?
(a) 30 x + 20 y
(b) (30 + 20) × (x + y)
(c) 20 x + 30 y
(d) (30 + 20) × x + y
(e) 30 x – 20 y
Answer:
(a) 30 x + 20 y, represents the expression describing the total amount in rupees earned that day.
2. Pushpita sells two types of flowers on Independence Day: champak and marigold. ‘p’ customers only bought champak, ‘ q ‘ customers only bought marigold, and ‘r’ customers bought both. On the same day, she gave away a tiny national flag to every customer. How many flags did she give away that day?
(a) p + q + r
(b) p + q + 2 r
(c) 2 × (p + q + r)
(d) p + q + r + 2
(e) p + q + r + 1
(f) 2 × (p + q)
Answer:
(b) p + q + 2r, represents the expression to find the number of flags given on that day.
3. A snail is trying to climb along the wall of a deep well. During the day it climbs up ‘u’ cm and during the night it slowly slips down ‘d’ cm. This happens for 10days and 10 nights.
(a) Write an expression describing how far away the snail is from its starting position.
(b) What can we say about the snail’s movement if d > u?
Answer:
distance climbed in 10days = 10 u
Distance slipped down in 10 nights = 10d
(a) An expression describing how far away the snail is from its starting position is given by:
10 u – 10d
(b) If d > u, then the snail will be below its starting point.
4. Radha is preparing for a cycling race and practices daily. The first week she cycles 5 km every day. Every week she increases the daily distance cycled by ‘ z ‘ km. How many kilometers would Radha have cycled after 3 weeks?
Answer:
In the first week, Radha & would have cycled 7 × 5 = 35 km.
She increases the daily distance by ‘ x ‘ km every week.
So in the second week she will cycle
7 × (5 + x) = (35 + 7 x) km
and in the third week she will cycle
7 ×(5 + 2 x) = (35 + 14 x) km
So, after 3 weeks, Radha would have cycled a total distance of :
35 + (35 + 7x) + (35 + 14 x)
= 35 + 35 + 7x + 35 + 14 x
= 105 + 21x
= (21x + 105) km.
5. In the following figure, observe how the expression w + 2 becomes 4w + 20 along one path. Fill in the missing blanks on the remaining paths. The ovals contain expressions and the boxes contain operations.
Answer:
6. A local train from Yahapur to Vahapur stops at three stations at equal distances along the way. The time taken in minutes to travel from one station to the next station is the same and is denoted by t. The train stops for 2 minutes at each of the three stations.
(a) If t = 4, what is the time taken to travel from Yahapur to Vahapur?
(b) What is the algebraic expression for the time taken to travel from Yahapur to Vahapur?
[Hint:draw a rough diagram to visualise the situation]
Answer:
(a) An arithmetic expression for the given situation is:
(4 + 2) + (4 + 2) + (4 + 2) + 4 = 6 + 6 + 6 + 4
= 22 minutes
(b) An algebraic expression for the time taken from Yahapur to Vahapur is:
t + 2 + t + 2 + t + 2 + t = 4 t + 6.
7. Simplify the following expressions:
(a) 3a + 9b – 6 + 8a – 4 b – 7 a + 16
(b) 3(3a – 3b) – 8a – 4 b – 16
(c) 2(2x – 3) + 8x + 12
(d) 8x – (2x – 3) + 12
(e) 8h – (5 + 7h) + 9
(f) 23 + 4(6m – 3n) – 8n – 3m – 18
Answer:
(a) 3a + 9b – 6 + 8a – 4b – 7a + 16
= [4a + 5 b + 10]
(b) 3(3 a – 3 b) – 8a – 4 b – 16
= 9 a – 9 b – 8 a – 4b – 16
= [a – 13 b – 16]
(c) 2(2x – 3) + 8 x + 12
= 4x – 6 + 8x + 12
= [12x + 6]
(d) 8x – (2x – 3) + 12
= 8x – 2x + 3 + 12
= [16 x + 15]
(e) 8h – (5 + 7 h) + 9
= 8 h – 5 – 7 h + 9
= [h + 4]
(f) 23 + 4(6m – 3n) – 8n – 3m – 18
= 23 + 24m – 12n – 8n – 3m – 18
= [21m – 21n + 5]
8. Add the expressions given below:
(a) 4d – 7c + 9 and 8c – 11 + 9d
(b) – 6f + 19 – 8s and – 23 + 13f + 12s
(c) 8d – 14c + 9 and 16c – (11 + 9d)
(d) 6f – 20 + 8s and 23 – 13f – 12s
(e) 13m – 12n and 12n – 13m
(f) -26m + 24n and 26m – 24n
Answer:
(a) 4d – 7c + 9 and 8c – 11 + 9d
or, c + 13d – 2
(b) – 6f + 19 – 8s and – 23 + 13f + 12s
(c) 8d – 14c + 9 and 16 c – (11 + 9d)
i.e., 8d – 14c + 9 and 16c – 11 – 9d
So,
or 2c – d – 2
(d) 6f – 20 + 8s and 23 – 13f – 12s
(e) 13m – 12n and 12n – 13m
So,
So, 0
(f) -26m + 24n and 26m – 24n
So,
9. Subtract the expressions given below:
(a) 9a – 6b + 14 from 6a + 9b – 18
(b) -15x + 13 – 9y from 7y – 10 + 3x
(c) 17g + 9 – 7h from 11 – 10g + 3h
(d) 9a – 6b + 14 from 6 a – (9b + 18)
(e) 10x + 2 + 10 y from – 3y + 8 – 3x
(f) 8g + 4h – 10 from 7h – 8g + 20
Answer:
(a) 9a – 6b + 14 from 6a + 9b – 18
So,
(b) -15x + 13 – 9y from 7y – 10 + 3x
So,
(c) 17g + 9 – 7 h from 11 – 10g + 3h
So,
(d) 9a – 6b + 14 from 6 a – (9b + 18)
So,
(e) 10x + 2 + 10y from – 3y + 8 – 3x
So,
(f) 8g + 4h – 10 from 7 h – 8g + 20
So,
10. Describe situations corresponding to the following algebraic expressions:
(a) 8x + 3y
(b) 15x – 2x
Answer:
(a) Aadya buys pens for ₹8 and y pencils for ₹3 each. The expression for the total amount to be paid is?
(b) Jethu purchased 15 notebooks for ₹ x each, but due to some defects in 2 of the notebooks purchased, he returned it to the shopkeeper. The expression for the amount to be paid by Jethu to the shopkeeper.
11. Imagine a straight rope. If it is cut once as shown in the picture, we get 2 pieces. If the rope is folded once and then cut as shown, we get 3 pieces. Observe the pattern and find the number of pieces if the rope is folded 10 times and cut. What is the expression for the number of pieces when the rope is folded r times and cut?
Answer:
Here, sequence of pieces obtained is: 3,4,5, … .
As for one time fold we have 3 pieces, two time fold we have 4 pieces, … .
So, for 10 times fold, we will have
10 + 2 pieces 12 pieces
Corresponding expression for the number of pieces when the rope is folded r times and cut is:
r + 2
12. Look at the matchstick pattern below. Observe and identify the pattern. How many matchsticks are required to make 10 such squares. How many are required to make w squares?
Answer:
For 1 square, number of matchsticks required
= 4 = 1 × 3 + 1
For 2 squares, number of matchsticks required
= 7 which is (4 + 3) = 2 × 3 + 1
For 3 squares, number of matchsticks required
= 10 = 3 × 3 + 1
So, for 10 such squares, number of matchsticks required.
= 10 × 3 + 1 = 31
Also, there will be 3w + 1 required for w squares.
13. Have you noticed how the colours change in a traffic signal? The sequence of colour changes is shown below.
Find the colour at positions 90, 190, and 343. Write expressions to describe the positions for each colour.
Answer:
The sequence for red colour: 1,5,9,13, ….. .
The sequence for yellow colour : 2, 4, 6, 8,…
The sequence for green colour : 3,7,11, …. .
At 90th and 190th positions the colour will be yellow as they are even positions.
Also, 343 comes in the sequence 3,7,11, …. .
So, at 343rd position it’s colour will be green.
For red colour the expression will be: 4n – 3
For yellow colour the expression will be: 2y,
For green colour the expression will be: 4z – 1
Note: By using above expressions, we can find the positions at which different colours will be there.
For example: To know the various positions at which green colour light will be seen we put z = 1,2,3,4,5, …. . to get 3,7,11,15,19, …. .
14. Observe the pattern below. How many squares will be there in Step 4, Step 10, Step 50? Write a general formula. How would the formula change if we want to count the number of vertices of all the squares?
Answer:
Formula here is : 4 × n + 1, where ‘ n ‘denotes the step.
We know that a square has 4 vertices. So, the formula for the number of vertices will be: 4 (4n + 1), where ‘ n ‘denotes the number of steps.
15. Numbers are written in a particular sequence in this endless 4 – column grid.
(a) Give expressions to generate all the numbers in a given column (1, 2, 3, 4).
(b) In which row and column will the following numbers appear:
(i) 124 (ii) 147 (iii) 201
(c) What number appears in row r and column c?
(d) Observe the positions of multiples of 3.
Do you see any pattern in it? List other patterns that you see.
Answer:
(a)
| Column | 4n-3 |
| 1 | 4m-2 |
| 2 | 4t-1 |
| 3 | 4u |
| 4 | 4n-3 |
Here, n, m, t and u denote the number of rows.
(b) (i) 124 will lie in the 4th column as 124 isdivisible by 4 and also
\(\frac{124}{4}\) = 31
So, 124 will lie in 31st row.
(ii) 147 will lie in the 3rd column as 147 + 1 = 148 which isdivisible by 4 and
\(\frac{148}{4}\) = 37
So, 147 will lie in 37th row.
(iii) 201 will lie in the I st Column as 201 + 3 is divisible by 4 and
\(\frac{204}{4}\) = 51
So, 201 will lie in 51st row.
(c) Expression for number appearing in row ‘ r ‘ and column ‘ c ‘ can be written as:
“4r + c – 4”
(d) Positions of multiples of 3 are:
In column 1: 3rd, 6th, 9th, …. . row
In column 2: 2nd, 5th, 8th, 11th, …. . row
In column 3: 1st, 4th, 7th, …. . row
In column 4: 3rd, 6th, 9th,… row
The pattern notices: They are coming after a gap of 3.
For other patterns:
Look for multiples of 5.
It is coming at:
| Column | Rows |
| 1 | 2, 7, 12, 17, … |
| 2 | 3, 8, 13, 18, … |
| 3 | 4, 9, 14, 19, … |
| 4 | 5, 10, 15, 20, … |