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

= 5^{3} × 2^{0}

∴ \(\frac{327}{125}\) is a terminating decimal

(ii) \(\frac{99}{800}\)

Answer:

800 = 32 × 25

= 2^{5} × 5^{2}

∴ \(\frac{99}{800}\) is a terminating decimal

(iii) \(\frac{17}{1250}\)

Answer:

1250 = 5^{4} × 2^{1}

∴ \(\frac{17}{1250}\) is a terminating decimal

(iv) \(\frac{29}{200}\)

Answer:

200 = 25 × 8

= 5^{2} × 2^{3}

∴ \(\frac{29}{200}\) 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 5^{n} × 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} × 2^{n}

∴ \(\frac{1276}{680}\) 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} × 2^{n}

∴ \(\frac{22}{190}\) 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} × 2^{n}

∴ \(\frac{11}{750}\) 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} × 5^{n}

∴ \(\frac{-17}{130}\) is a non-terminating decimal.

(ii) \(\frac{21}{128}\)

Answer:

\(\frac{21}{128}\), 128 = 2 × 2 × 2 × 2 × 2 × 2

= 2^{7} × 5^{n}

∴ \(\frac{21}{128}\) is a terminating decimal.

(iii) \(\frac{35}{216}\)

Answer:

\(\frac{35}{216}\), 216 = 2 × 2 × 2 × 3 × 3 × 3

= 2^{3} × 3^{3}

∴ \(\frac{35}{216}\) is a not a terminating decimal.

(iv) \(\frac{31}{200}\)

Answer:

\(\frac{31}{200}\), 200 = 2 × 2 × 2 × 5 × 5 = 2^{3} × 5^{2}

Here the prime factors of 200 are in the form of 2^{m} × 5^{n}

∴ 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