NCERT Exemplar Solutions
Class 7 - Mathematics

Unit 8: Rational Numbers

\(-\dfrac{3}{8}\) is a _____ rational number.

1 is a _____ rational number.

The standard form of \(-\dfrac{8}{-36}\) is _____.

The standard form of \(\dfrac{18}{-24}\) is _____.

On a number line, \(-\dfrac{1}{2}\) is to the _____ of zero (0).

On a number line, \(\dfrac{4}{3}\) is to the _____ of zero (0).

\(-\dfrac{1}{2}\) is _____ than \(\dfrac{1}{5}\).

\(-\dfrac{3}{5}\) is _____ than 0.

\(-\dfrac{16}{24}\) and \(\dfrac{20}{-16}\) represent _____ rational numbers.

\(-\dfrac{27}{45}\) and \(-\dfrac{3}{5}\) represent _____ rational numbers.

Additive inverse of \(\dfrac{2}{3}\) is _____.

\(-\dfrac{3}{5} + \dfrac{2}{5} = \) _____.

\(-\dfrac{5}{6} + -\dfrac{1}{6} = \) _____.

\(\dfrac{3}{4} \times ( -\dfrac{2}{3} ) = \) _____.

\(-\dfrac{5}{3} \times ( -\dfrac{3}{5} ) = \) _____.

\(-\dfrac{6}{7} = \dfrac{__}{42}\)

\(\dfrac{1}{2} = \dfrac{6}{__}\)

\(-\dfrac{2}{9} - \dfrac{7}{9} = \) _____.

\(-\dfrac{7}{8} \; \Box \; \dfrac{8}{9}\)

\(\dfrac{3}{7} \; \Box \; \dfrac{-5}{6}\)

\(\dfrac{5}{6} \; \Box \; \dfrac{8}{4}\)

\(-\dfrac{9}{7} \; \Box \; \dfrac{4}{-7}\)

\(\dfrac{8}{8} \; \Box \; \dfrac{2}{2}\)

The reciprocal of _____ does not exist.

The reciprocal of 1 is _____.

\(-\dfrac{3}{7} \div ( -\dfrac{7}{3} ) = \) _____

\(0 \div ( -\dfrac{5}{6} ) = \) _____

\(0 \times ( -\dfrac{5}{6} ) = \) _____

_____ \(\times ( -\dfrac{2}{5} ) = 1\)

The standard form of rational number -1 is _____.

If m is a common divisor of a and b, then \(\dfrac{a}{b} = \dfrac{a \div m}{\_\_\_}\)

If p and q are positive integers, then \(\dfrac{p}{q}\) is a _____ rational number and \(\dfrac{p}{-q}\) is a _____ rational number.

Two rational numbers are said to be equivalent or equal, if they have the same _____ form.

If \(\dfrac{p}{q}\) is a rational number, then q cannot be _____.

Every natural number is a rational number but every rational number need not be a natural number.

Zero is a rational number.

Every integer is a rational number but every rational number need not be an integer.

Every negative integer is not a negative rational number.

If \(\dfrac{p}{q}\) is a rational number and m is a non-zero integer, then \(\dfrac{p}{q} = \dfrac{p \times m}{q \times m}\).

If \(\dfrac{p}{q}\) is a rational number and m is a non-zero common divisor of p and q, then \(\dfrac{p}{q} = \dfrac{p \div m}{q \div m}\).

In a rational number, denominator always has to be a non-zero integer.

If \(\dfrac{p}{q}\) is a rational number and m is a non-zero integer, then \(\dfrac{p \times m}{q \times m}\) is a rational number not equivalent to \(\dfrac{p}{q}\).

Sum of two rational numbers is always a rational number.

All decimal numbers are also rational numbers.

The quotient of two rationals is always a rational number.

Every fraction is a rational number.

Two rationals with different numerators can never be equal.

8 can be written as a rational number with any integer as denominator.

\(\dfrac{4}{6}\) is equivalent to \(\dfrac{2}{3}\).

The rational number \(-\dfrac{3}{4}\) lies to the right of zero on the number line.

The rational numbers \(-\dfrac{12}{-5}\) and \(-\dfrac{7}{17}\) are on the opposite sides of zero on the number line.

Every rational number is a whole number.

Zero is the smallest rational number.

66. Match the following:

Write each of the following rational numbers with positive denominators: \(\dfrac{5}{-8}, \dfrac{15}{-28}, \dfrac{-17}{-13}\).

Express \(\dfrac{3}{4}\) as a rational number with denominator:

(i) 36

(ii) -80

Reduce each of the following rational numbers in its lowest form:

(i) \(\dfrac{-60}{72}\)

(ii) \(\dfrac{91}{-364}\)

Express each of the following rational numbers in its standard form:

(i) \(\dfrac{-12}{-30}\)

(ii) \(\dfrac{14}{-49}\)

(iii) \(\dfrac{-15}{35}\)

(iv) \(\dfrac{299}{-161}\)

Are the rational numbers \(\dfrac{-8}{28}\) and \(\dfrac{32}{-112}\) equivalent? Give reason.

Arrange the rational numbers \(-\dfrac{7}{10}, \dfrac{5}{-8}, \dfrac{2}{-3}, -\dfrac{1}{4}, -\dfrac{3}{5}\) in ascending order.

Represent the following rational numbers on a number line: \(\dfrac{3}{8}, -\dfrac{7}{3}, \dfrac{22}{-6}\).

If \(-\dfrac{5}{7} = \dfrac{x}{28}\), find the value of x.

Give three rational numbers equivalent to:

(i) \(-\dfrac{3}{4}\)

(ii) \(\dfrac{7}{11}\)

Write the next three rational numbers to complete the pattern:

(i) \(\dfrac{4}{-5}, \dfrac{8}{-10}, \dfrac{12}{-15}, \dfrac{16}{-20}, \_\_, \_\_, \_\_\)

(ii) \(\dfrac{-8}{7}, \dfrac{-16}{14}, \dfrac{-24}{21}, \dfrac{-32}{28}, \_\_, \_\_, \_\_\)

List four rational numbers between \(\dfrac{5}{7}\) and \(\dfrac{7}{8}\).

Find the sum of:

(i) \(\dfrac{8}{13} + \dfrac{3}{11}\)

(ii) \(\dfrac{7}{3} + \dfrac{-4}{3}\)

Solve:

(i) \(\dfrac{29}{4} - \dfrac{30}{7}\)

(ii) \(\dfrac{5}{13} - \dfrac{-8}{26}\)

Find the product of:

(i) \(\dfrac{-4}{5} \times \dfrac{-5}{12}\)

(ii) \(\dfrac{-22}{11} \times \dfrac{-21}{11}\)

Simplify:

(i) \(\dfrac{13}{11} \times \dfrac{-14}{5} + \dfrac{13}{11} \times \dfrac{-7}{5} + \dfrac{-13}{11} \times \dfrac{34}{5}\)

(ii) \(\dfrac{6}{5} \times \dfrac{3}{7} - \dfrac{1}{5} \times \dfrac{3}{7}\)

Simplify:

(i) \(\dfrac{3}{7} \div (\dfrac{21}{-55})\)

(ii) \(1 \div ( -\dfrac{1}{2} )\)

Which is greater in the following?

(i) \(\dfrac{3}{4}, \dfrac{7}{8}\)

(ii) -\(\dfrac{3}{5}, \dfrac{1}{9}\)

Write a rational number in which the numerator is less than \(-7×11\) and the denominator is greater than \(12+4\).

If \(x=\dfrac{1}{10}\) and \(y=\dfrac{-3}{8}\), then evaluate \(x+y, x-y, x×y, x÷y\).

Find the reciprocal of the following:

(i) \((\dfrac{1}{2} \times \dfrac{1}{4}) + (\dfrac{1}{2} \times 6)\)

(ii) \(\dfrac{20}{51} \times \dfrac{4}{91}\)

(iii) \(\dfrac{3}{13} \div \dfrac{-4}{65}\)

(iv) \((-5 \times \dfrac{12}{15}) - (-3 \times \dfrac{2}{9})\)

Complete the following table by finding the sums:

Write each of the following numbers in the form \(\dfrac{p}{q}\):

(a) six-eighths

(b) three and half

(c) opposite of 1

(d) one-fourth

(e) zero

(f) opposite of three-fifths

If \(p = m \times t\) and \(q = n \times t\), then \(\dfrac{p}{q} = \_\_\).

Given \(\dfrac{p}{q}\) and \(\dfrac{r}{s}\) are rationals in standard form:

(a) \(\dfrac{p}{q} < \dfrac{r}{s}\), if \(p \times s < r \times q\)

(b) \(\dfrac{p}{q} = \dfrac{r}{s}\), if \(p \times s = r \times q\)

(c) \(\dfrac{p}{q} > \dfrac{r}{s}\), if \(p \times s > r \times q\)

In each of the following cases, write the rational number whose numerator and denominator are respectively as under:

(a) 5–39 and 54–6

(b) (-4)×6 and 8÷2

(c) 35÷(-7) and 35–18

(d) 25+15 and 81÷40

Write the following as rational numbers in their standard forms:

(a) 35%

(b) 1.2

(c) -6 3/7

(d) 240 ÷ (-840)

(e) 115 ÷ 207

Find a rational number exactly halfway between:

(a) -1/3 and 1/3

(b) 1/6 and 1/9

(c) 5/-13 and -7/9

(d) 1/15 and 1/12

Taking x=-4/9, y=5/12, z=7/18, find:

(a) rational number added to x gives y

(b) rational number subtracted from y gives z

(c) rational number added to z gives x

(d) rational number multiplied by y gives x

(e) reciprocal of x+y

(f) sum of reciprocals of x and y

(g) (x÷y)×z

(h) (x–y)+z

(i) x+(y+z)

(j) x ÷ (y ÷ z)

(k) x–(y+z)

What should be added to -1/2 to obtain the nearest natural number?

What should be subtracted from -2/3 to obtain the nearest integer?

What should be multiplied with -5/8 to obtain the nearest integer?

What should be divided by 1/2 to obtain the greatest negative integer?

A rope 68m long is cut into pieces of length 4 1/4 m. Find number of such pieces.

If 12 shirts of equal size are prepared from 27m cloth, what is length required for each shirt?

Insert 3 equivalent rational numbers between:

(i) -1/2 and 1/5

(ii) 0 and -10

Put (√) wherever applicable:

a and b are different numbers from 1–50. What is largest value of (a–b)/(a+b) and (a+b)/(a–b)?

150 students study English, Maths or both. 62% study English, 68% study Maths. How many study both?

A body floats 2/9 above surface. What is ratio of submerged volume to exposed volume?

Find the odd one out:

(a) 4/3×3/4

(b) -3/2×-2/3

(c) 2×1/2

(d) -1/3×3/1

Find the odd one out:

(a) 4/-9

(b) -16/36

(c) -20/-45

(d) 28/-63