DAV Class 7 Maths Chapter 3 Brain Teasers Solutions

The DAV Class 7 Maths Book Solutions Pdf and DAV Class 7 Maths Chapter 3 Brain Teasers Solutions of Rational Numbers as Decimals offer comprehensive answers to textbook questions.

DAV Class 7 Maths Ch 3 Brain Teasers Solutions

Question 1.
A. Tick (✓) the correct option:
(i) 0.225 expressed as a rational number is
(a) \(\frac{1}{4}\)
(b) \(\frac{45}{210}\)
(c) \(\frac{9}{40}\)
(d) \(\frac{225}{999}\)
Answer:
(c) \(\frac{9}{40}\)

0.225 = \(\frac{225}{1000}=\frac{9}{40}\)
Hence, (c) is the correct answer.

(ii) A rational number \(\frac{p}{q}\) can be expressed as a terminating decimal if q ha no factor other than:
(a) 2, 3
(b) 2, 5
(c) 3, 5
(d) 2, 3, 5
Answer:
(b) 2, 5

If the denominator (q) of a rational number (\(\frac{p}{q}\)) is the standard form has 2 or 5 or both only prime factors, then it can be represented as a terminating decimal
Hence, (b) is the correct answer

(iii) -7 \(\frac{8}{100}\) expressed as a decimal number is:
(a) -7.800
(b) 7.008
(c) -7.008
(d) -7.08
Answer:
(d) -7.08

-7\(\frac{8}{100}=\frac{-708}{100}\) = -7.08
Hence, (d) is the correct answer.

(iv) 4.013\(\overline{25}\) is equal to
(a) 4.013252525…….
(b) 4.01325555….
(c) 4.0132501325……
(d) 4.0130132525……..
Answer:
(a) 4.013252525…….

4.013\(\overline{25}\) = 4.013252525…….
Hence, (a) is the correct answer.

(v) The quotient when 0.00639 is divided by
(a) 3
(b) 0.3
(c) 0.03
(d) 0.003
Answer:
(c) 0.03

\(\frac{0.00639}{0.213}=\frac{639}{100000} \times \frac{1000}{213}\)
= \(\frac{3}{100}\) = 0.03
Hence, (c) is the correct answer.

B. Answer the following questions:
(i) Without actual division, determine if is terminating or non-terminating decimal number.
Answer:
The denominator of \(\frac{-28}{250}\) is 250.
250 = 2 × 5 × 5 × 5.
The prime factors of 250 are 2 and 5. But, the rational number is not in its lowest form.
In fact, \(\frac{-28}{250}=\frac{-14 \times 2}{125 \times 2}=\frac{-14}{125}\), whose denominator 125 = 5 x 5 x 5
Therefore, \(\frac{-28}{250}\) has a terminating decimal representation.

(ii) Convert \(\frac{-113}{7}\) to decimals.
Answer:
To represent \(\frac{-113}{7}\) as a decimal, we divide 113 by 7.

∴ \(\frac{-113}{7}\) = 16.142857…… = 16.\(\overline{142857}\)
\(\frac{-113}{7}\) in decimal form is represented as 16.\(\overline{142857}\)
Therefore, \(\frac{-113}{7}\) = 16.\(\overline{142857}\)

(iii) What should be subtracted from -15.834 to get 3.476?
Answer:
Let x is subtracted from -15.834 to get 3.476
∴ -15.834 – x = 3.476
⇒ x = -15.834 – 3.476
⇒ x = -19.310
Therefore, -19.310 should be subtracted from – 15.834 to get 3.476

(iv) Express 4.82 as rational number in standard form.
Answer:
4.82 = \(\frac{482}{100}=\frac{241}{50}\)

(v) Find the value of 16.016 ÷ 0.4.
Answer:
16.016 ÷ 0.4
= \(\frac{16.016}{0.4}=\frac{16016}{1000} \times \frac{10}{4}=\frac{4004}{100}\) = 40.04

Question 2.
Convert the following rational numbers into decimals:
(i) \(\frac{259}{3}\)
Answer:

∴ \(\frac{259}{3}\) = 86.333 … = 86.3̄

(ii) \(\frac{19256}{11}\)
Answer:

∴ \(\frac{19256}{11}\) = 175.5454…….
= 175.\(\overline{54}\)

(iii) \(\frac{15735}{80}\)
Answer:

(iv) \(\frac{27}{7}\)
Answer:

∴ \(\frac{27}{7}\) = 3.857142857142 = 3.\(\overline{857142}\)

(v) \(\frac{758}{1250}\)
Answer:

(vi) \(\frac{15625}{12}\)
Answer:

Question 3.
Find the decimal representation of the following rational numbers:
(i) \(\frac{-12}{13}\)
Answer:

∴ \(\frac{-12}{13}\) = -0.923076923076….. = -0.\(\overline{923076}\)

(ii) \(\frac{-1525}{50}\)
Answer:

(iii) \(\frac{-127}{7}\)
Answer:

Hence \(\frac{-127}{7}\) = -18.142857142857…… = -18.\(\overline{142857}\)

(iv) \(-\frac{539}{80}\)
Answer:

∴ Hence \(-\frac{539}{80}\) = -6.7375

Question 4.
Simplify the following expressions:
(i) 3.2 + 16.09 + 26.305 – 1.232
Answer:
3.2 + 16.09 + 26.305 -1.232
= 3.200 +16.090 + 26.305 -1.232
= 45.595-1.232
= 44.363

(ii) -5.7 + 13.20 – 15.009 + 0.02
Answer:
-5.7 + 13.20-15.009 + 0.02
= – 5.700 + 13.200 – 15.009 + 0.020
= (13.200 – 5.700) + (0 .020 -15.009)
= 7.500 + (-14.989)
= 7.500-14.989
= -7.489

(iii) (0.357 + 0.96) – (3.25 – 2.79)
Answer:
(0.357 + 0.96) – (3.25 – 2.79)
= (0.357 + 0.960) – (3.250 – 2.790)
= 1.317-0.460
= 0.857

(iv) 15 + 2.57 – 23.07 – 5.003
Answer:
15 + 2.57 – 23.07 – 5.003
= 15.000 + 2.570 – 23.070 – 5.003
= 17.570 – 28.073
= -10.503

Question 5.
Without actual division, determine which of the following rational numbers have a terminating decimal representation.
(i) \(\frac{327}{125}\)
Answer:
125 = 5 × 5 × 5
= 5³ × 2⁰
∴ \(\frac{327}{125}\) is a terminating decimal

(ii) \(\frac{99}{800}\)
Answer:
800 = 32 × 25
= 2⁵ × 5²
∴ \(\frac{99}{800}\) is a terminating decimal

(iii) \(\frac{17}{1250}\)
Answer:
1250 = 5⁴ × 2¹
∴ \(\frac{17}{1250}\) is a terminating decimal

(iv) \(\frac{29}{200}\)
Answer:
200 = 25 × 8
= 5² × 2³
∴ \(\frac{29}{200}\) is a terminating decimal

(v) \(\frac{135}{1625}\)
Answer:
\(\frac{135}{1625}=\frac{135 \div 5}{1625+5}=\frac{27}{325}\)
325 = 5 × 5 × 13
It is not in the form of 5ⁿ × 2^m
∴ \(\frac{135}{1625}\) is a non – terminating decimal

(vi) \(\frac{1276}{680}\)
Answer:
\(\frac{1276}{680}=\frac{1276 \div 4}{680 \div 4}=\frac{319}{170}\)
170 = 2 × 5 × 17
It is not in the form of 5^m × 2ⁿ
∴ \(\frac{1276}{680}\) is a non – terminating decimal

(vii) \(\frac{22}{190}\)
Answer:
\(\frac{22}{190}=\frac{22 \div 2}{190 \div 2}=\frac{11}{95}\)
95 = 5 × 19
It is not in the form of 5^m × 2ⁿ
∴ \(\frac{22}{190}\) is a non – terminating decimal

(viii) \(\frac{11}{750}\)
Answer:
750 = 3 × 5 × 5 × 5 × 2
It is not in the form of 5^m × 2ⁿ
∴ \(\frac{11}{750}\) is a non – terminating decimal

Question 6.
Simplify the following and express the result as decimals.
(i) 2.7 × 1.5 × 2.1
Answer:

2.7 × 1.5 × 2.1
= 4.05 × 2.1
= 8.505

(ii) 12 × 13.6 × 0.25
Answer:

12 × 13.6 × 0.25
= 163. 2 × 0.25
= 40.800
= 40.8

(iii) 3.25 × 72.6
Answer:

3.25 × 72.6
= 231.95

(iv) (156.25 ÷ 0.0251 × 0.02 – 5.2
Answer:
(156.25 ÷ 0.025) × 0.02 – 5.2
= \(\frac{156.25}{0.025}\) × 0.02 – 5.2
= \(\frac{156250}{25}\) × 0.02 – 5.20
= 6250 × 0.02 – 5.20
= 125.00 – 5.20
= 125 – 5.20
= 119.80
= 119.8

(v) (75.05 ÷ 0.05) × 0.001 + 2.351
Answer:
(75.05 ÷ 0.05) × 0.001 + 2.351
= \(\frac{75.05}{0.05}\) × 0.001 + 2.351
= \(\frac{7505}{5}\) × 0.001 + 2.351
= 1501 × 0.001 + 2.351
= 1.501 + 2.351
= 3.852

Question 7.
Simplify and express the result as a rational number in its lowest form,
(i) 3.125 + 0.125 + 0.50 – 0.225
Answer:
3.125 + 0.125 + 0.50 – 0.225

(ii) \(\frac{0.4 \times 0.04 \times 0.005}{0.1 \times 10 \times 0.001}-\frac{1}{2}+\frac{1}{5}\)
Answer:

(iii) \(\frac{0.144 \div 1.2}{0.016 \div 0.02}+\frac{7}{5}-\frac{21}{8}\)
Answer:

DAV Class 7 Maths Chapter 3 HOTS

Question 1.
Perimeter of a rectangle is 2.4 m less than \(\frac{2}{5}\) of the perimeter of a square. If the perimeter of the square is 40 m, find the length and breadth of the rectangle give that breadth is \(\frac{1}{3}\) of the length.
Answer:
Let the length and breadth of the rectangle be l and b respectively.
∴ Perimeter of rectangle = 2(1 + b)
According to the question,
2(l + b) = \(\frac{2}{5}\) (40) – 2.4
⇒ 2(l + b) = 2(8) – 2.4
⇒ 2(l + b) = 16 – 2.4
⇒ 2(l + b) = 13.6 …(1)

Also, b = \(\frac{1}{3}\)l [Given]
⇒ 3 b = l
⇒ l = 3b
Putting l = 3b in equation (1), we get
2(3 b + b) = 13.6
⇒ 2(4 b) = 13.6
⇒ 8b = 13.6
⇒ b = \(\frac{13.6}{8}\)
⇒ b = 1.7 m
∴ l = 3b
⇒ l = 3 × 1.7 m
= 5.1 m
Therefore, the length and breadth of the rectangle are 5.1 m and 1.7 m respectively.

DAV Class 7 Maths Chapter 3 Enrichment Questions

Question 1.
There is an interesting pattern in the following:

You will question why the left hand side in each case is 1, but, the right hand side is 0.9 ? (You will learn about this in higher classes).
Notice they all have only the digits 142857, each starting with a different digit but in the same order.
Try finding out the repeating part of the decimal for \(\frac{1}{13}\). What do you notice?
Answer:
To represent \(\frac{1}{13}\) asa decimal, we divide 1 by 13.

∴ \(\frac{1}{13}\) = 0.076923… = 0.\(\overline{076923}\)
Here, we have remainder as 1 which is just the same as the dividend. Therefore, after 3, the same digits, i.e., 0, 7, 6, 9, 2, 3 will keep on repeating again and again in the quotient.

Additional Questions

Question 1.
Convert the following in the decimal form.
(i) \(\frac{-3}{125}\)
Answer:
\(\frac{-3}{125}=\frac{-3 \times 8}{125 \times 8}\)
= \(\frac{-24}{1000}\)
= -0.024

(ii) \(\frac{4}{25}\)
Answer:
\(\frac{4}{25}=\frac{4 \times 4}{25 \times 4}\)
= \(\frac{16}{100}\)
= -0.16

(iii) \(\frac{-7}{80}\)
Answer:
\(\frac{-7}{80}=\frac{-7 \times 125}{80 \times 125}\)
= \(\frac{-875}{10000}\)
= -0.0875

(iv) \(\frac{-12}{5}\)
Answer:
\(\frac{-12}{5}=\frac{-12 \times 2}{5 \times 2}\)
= \(\frac{-24}{10}\)
= -2.4

Question 2.
Express the following rational numbers as decimals by using long division method.
(i) \(\frac{629}{125}\)
Answer:

∴ \(\frac{629}{125}\) = 5.032

(ii) \(\frac{317}{80}\)
Answer:

∴ \(\frac{317}{80}\) = 3.9625

Question 3.
Express the following rational numbers as decimals by using long division method.
(i) \(\frac{-23}{7}\)
Answer:

∴ \(\frac{-23}{7}\) = -3.285714285714 ………… = -3.\(\overline{285714}\)

(ii) \(\frac{-37}{4}\)
Answer:

∴ \(\frac{-37}{4}\) = 9.25

(iii) \(\frac{62}{3}\)
Answer:

∴ \(\frac{62}{3}\) = 20.6̄

(iv) \(\frac{523}{15}\)
Answer:

∴ \(\frac{523}{15}\) = 34.866.. = 34.86̄

Question 4.
Without actual division find which of the following rational numbers have terminating decimal representation?
(i) \(\frac{-17}{130}\)
Answer:
\(\frac{-17}{130}\), 130 = 2 × 5 × 13
Here prime factors are not in the form 2^m × 5ⁿ
∴ \(\frac{-17}{130}\) is a non-terminating decimal.

(ii) \(\frac{21}{128}\)
Answer:
\(\frac{21}{128}\), 128 = 2 × 2 × 2 × 2 × 2 × 2
= 2⁷ × 5ⁿ
∴ \(\frac{21}{128}\) is a terminating decimal.

(iii) \(\frac{35}{216}\)
Answer:
\(\frac{35}{216}\), 216 = 2 × 2 × 2 × 3 × 3 × 3
= 2³ × 3³
∴ \(\frac{35}{216}\) is a not a terminating decimal.

(iv) \(\frac{31}{200}\)
Answer:
\(\frac{31}{200}\), 200 = 2 × 2 × 2 × 5 × 5 = 2³ × 5²
Here the prime factors of 200 are in the form of 2^m × 5ⁿ
∴ It is a terminating decimal.

Question 5.
Express the following rational numbers in the form of \(\frac{p}{q}\).
(i) 0.08
Answer:
0.08 = \(\frac{8}{100}=\frac{8 \div 4}{100 \div 4}\)
= \(\frac{2}{25}\)

(ii) -0.16
Answer:
-0.16 = \(\frac{-16}{100}=\frac{-16 \div 4}{100 \div 4}\)
= \(\frac{-4}{25}\)

(iii) -24.25
Answer:
-24.25 = \(\frac{-2425}{100}=\frac{-2425 \div 25}{100 \div 25}\)
= \(\frac{-97}{4}\)

(iv) -0.25
Answer:
-0.25 = \(\frac{-25}{100}=\frac{-25 \div 25}{100 \div 25}\)
= \(\frac{-1}{4}\)

Question 6.
Simplify: \(\frac{2}{3}+\frac{31}{48}-\frac{16}{9}+\frac{1}{12}\)
Answer:
L.C.M of 3, 48, 9, 12

= 2 × 2 × 3 × 3 × 2 × 2 = 144

Question 7.
Evaluate: 215.3 – 14.003 + 0.02 – 0.005
Answer:
215.3 – 14.003 + 0.02 – 0.005
= 215.300 – 14.003 + 0.020 – 0.005
= 215.300 + 0.020 – 14.003 – 0.005
= 215.320 -14.008
= 201.312

Question 8.
Simplify the following and express the result in standard form.
(i) \(\frac{3}{4} \div \frac{7}{8} \times \frac{5}{16}+\frac{4}{5}-\frac{1}{3}\)
Answer:

(ii) \(\frac{0.001 \times 0.12}{3.50 \times 0.36}\)
Answer:
\(\frac{0.001 \times 0.12}{3.50 \times 0.36}=\frac{0.00012}{1.2600}\)
⇒ \(\frac{0.00012}{1.26000}=\frac{12}{126000}\)
⇒ \(\frac{12 \div 12}{126000 \div 12}=\frac{1}{10500}\)

Question 9.
Simplify the following expressions.
(i) (75.05 + 0.05) × 0.015 + 6.023
Answer:
(75.05 + 0.05) × 0.015 + 6.023

= 1501 × 0.015 + 6.023
= 22515 + 6.023
= 28.538

(ii) (30.05 + 0.01) + 0.005 + 2.005
Answer:

= 601000 + 2.005
= 601002.005

Question 10.
Evaluate the following:
(i) 43.7 – 13 – 10.025 + 3.18
Answer:
43.7 + 3.18 – 13 – 10.025
= 43. 700 + 3.180 -13.000 -10.025
= 46.880-23.025
= 23.855

(ii) (8.05 + 6.23) – (3.75 – 0.05)
Answer:
(8.05 + 6.23) – (3.75 – 0.05)
= 14.28 – 3.70
= 10.58