Rational Numbers(7th Grade > Mathematics ) Questions and answers for exam Preparation

7th Grade > Mathematics

RATIONAL NUMBERS MCQs

Total Questions : 120 | Page 5 of 12 pages

Question 41. (a) Amit used to be on the playground

4 \frac{1}{2}

times as much as his brother. Now he cut down

1 \frac{1}{3}

of the time he used to be on the playground as much as his brother. How much time does he spend on the playground when compared to his brother?
(b) The sum of two rational numbers is

\frac{- 5}{8}

. If one of the numbers is

\frac{- 6}{11}

, find the other number.
(c) Find x such that

\frac{- 3}{8}

and

\frac{x}{- 24}

are equivalent fractions.
[3 MARKS]

Discuss Question

:
Each part: 1 Mark
(a) Amit played

4 \frac{1}{2} = \frac{9}{2}

times of his brother.
He cut down his play by

1 \frac{1}{3} = \frac{4}{3}

The duration Amit plays now
=

\frac{9}{2} - \frac{4}{3} = \frac{19}{6} = 3 \frac{1}{6}

.
So,now Amit plays

3 \frac{1}{6}

times as his brother.
(b) Let the number be x.

\frac{- 6}{11} + x = \frac{- 5}{8}

x = \frac{- 5}{8} - \frac{- 6}{11}

x = \frac{- 5}{8} + \frac{6}{11}

x = \frac{- 7}{88}

\frac{- 3}{8}

and

\frac{x}{- 24}

to be equivalent.

\Rightarrow \frac{- 3}{8} = \frac{x}{- 24}

\Rightarrow x = \frac{- 3 \times - 24}{8}

\Rightarrow x = \frac{- 3 \times - 3}{1} = 9

Question 42. (a)

(a)A, B, C And D Are Rectangles. The Length Of The Diagonal...

A, B, C and D are rectangles.
The length of the diagonal of which rectangles are not irrational and which are irrational?
(b) Solve:

\frac{5}{13} - \frac{- 8}{26}

[4 MARKS]

Discuss Question

:
(a) Formula: 1 Mark
Steps: 1 Mark
Answer: 1 Mark
(b) Answer: 1 Mark
(a) To find the length of the diagonal, d, of each rectangle, use PythagorasTheorem.
For A, d

= \sqrt{(} 1^{2} + 2^{2}) = \sqrt{(} 1 + 4) = \sqrt{(} 5)

For B, d

= \sqrt{(} 2^{2} + 3^{2}) = \sqrt{(} 4 + 9) = \sqrt{(} 13)

For C, d

= \sqrt{(} 3^{2} + 4^{2}) = \sqrt{(} 9 + 16) = \sqrt{(} 25)

For D, d

= \sqrt{(} 4^{2} + 5^{2}) = \sqrt{(} 16 + 25) = \sqrt{(} 41)

Of these, only

\sqrt{(} 25) = 5

is rational.
The length of the diagonal of rectangle C is rational.
The length of the diagonal of rectangle A, Band D are irrational
(b)

\frac{5}{13} - \frac{- 8}{26}

\Rightarrow \frac{5}{13} + \frac{8}{26}

\Rightarrow \frac{10 + 8}{26}

\Rightarrow \frac{18}{26} = \frac{9}{13}

Question 43. (a) Ram is training for his school's annual sports meet. He earlier used to run at 8 m/s, now after training, he has increased his speed by

\frac{1}{2}

times his original speed. Find how much time will Ram take to complete a 100m race.
(b) List four rational numbers that lie between

\frac{5}{7}

and

\frac{7}{8}

[4 MARKS]

Discuss Question

:
Each part: 2 Marks
(a) Original speed of Ram

= 8

m/s
After training, his speed has increased to =

8 + \frac{1}{2} \times 8 = 12

m/s
Time taken to complete the 100m race

= \frac{100}{12} = \frac{25}{3}

seconds.
(b) One of the methods is taking the average of the two rational numbers. Let's do it with LCM method here.
Taking LCM of 7 and 8 = 56

\Rightarrow \frac{5}{7} = \frac{5 \times 8}{7 \times 8} = \frac{40}{56}

\Rightarrow \frac{7}{8} = \frac{7 \times 7}{8 \times 7} = \frac{49}{56}

∴

Four rational numbers between

\frac{40}{56}

and

\frac{49}{56}

can be

\frac{43}{56}, \frac{44}{56}, \frac{45}{56}, \frac{46}{56}

In standard form the rational numbers are

\frac{43}{56}, \frac{11}{14}, \frac{45}{56}, \frac{23}{28}

Question 44. Define Rational Number. [1 MARK]

Discuss Question

:
Any number which can be represented in the form of

\frac{p}{q}

where

p

and

q

are integers and

q \neq 0

is a rational number.

Question 45. (a) How many rational numbers lie between

\frac{2}{5}

and

\frac{6}{5}

? Mention atleast two.
(b) If

x = \frac{1}{10}

y = \frac{- 3}{8}

. Evaluate:

(x + y), (x - y), (x \times y), (x \div y)

[3 MARKS]

Discuss Question

:
a) Each rational number: 0.5 Marks
b) Each operation: 0.5 Marks
(a) There are infinite numbers of rational numbers that lie between two points on a number line.
Consider the rational numbers

\frac{2}{5}

and

\frac{6}{5}

.
Take the average of two numbers

\frac{2 + 6}{5 \times 2}

\frac{8}{10}

\frac{8}{10}

lies in the middle of

\frac{2}{5}

and

\frac{6}{5}

.
Now take the average of

\frac{2}{5}

and

\frac{8}{10}

. It is

(\frac{2}{5} + \frac{8}{10}) \times \frac{1}{2}

\frac{4 + 8}{10} \times \frac{1}{2}

\frac{12}{20}

So, as we proceed in this, we will get an infinite number of rational number between two points.
(b) (i)

x + y = \frac{1}{10} + \frac{- 3}{8}

\Rightarrow \frac{1}{10} - \frac{3}{8} = \frac{8 - 30}{80}

\Rightarrow \frac{- 22}{80} = \frac{- 11}{40}

(ii)

x - y = \frac{1}{10} - \frac{- 3}{8}

\Rightarrow \frac{1}{10} + \frac{3}{8} = \frac{8 + 30}{80}

\Rightarrow \frac{38}{80} = \frac{19}{40}

(iii)

x \times y = \frac{1}{10} \times \frac{- 3}{8} = \frac{- 3}{80}

(iv)

x \div y = \frac{1}{10} \div \frac{- 3}{8}

\Rightarrow \frac{1}{10} \times \frac{8}{- 3} = \frac{8}{- 30}

\Rightarrow \frac{- 8}{30} = \frac{- 4}{15}

Question 46. Give the additive inverse of the reciprocal of: [2 MARKS]
(i)

- \frac{2}{5}

(ii)

- \frac{17}{16}

Discuss Question

:
Each part: 1 Mark
(i) Reciprocal of

- \frac{2}{5} = - \frac{5}{2}

Additive Inverse of

- \frac{5}{2} = \frac{5}{2}

(ii) Reciprocal of

- \frac{17}{16} = - \frac{16}{17}

Additive Inverse of

- \frac{16}{17} = \frac{16}{17}

Question 47. (a) A ball is dropped from a height of 30m. After striking the surface it rises to

\frac{2}{3}

of its height. Again it falls on the surface and this time it covered only

\frac{2}{5}

of its previous height. It continued for next two times and it only covered half of its previous height. Find the distance covered by the ball.
(b) Which of the following inequalities are correct?
(i)

\frac{- 1}{5} < \frac{- 3}{5}

(ii)

\frac{- 3}{5} < \frac{- 1}{5}

(iii)

\frac{3}{5} > \frac{1}{5}

[4 MARKS]

Discuss Question

:
Each part: 2 Marks
(a) After striking to the surface, theheight gained by ball is

30 m \times \frac{2}{3} = 20 m

.
Now it attained the height of 20m, height attained by ball after

2^{n d}

strike is

20 \times \frac{2}{5} = 8 m

.
After

2^{n d}

strike, height attained by ball is

8 \times \frac{1}{2} = 4

.
So for

3^{r d}

strike, it falls from a height of 4m.
Again he attained half of the height of previous so this time it covers 2m.
So total distance covered by ball = 30+ 20+8+4+2 = 64.
(b)(i)

\frac{- 1}{5} < \frac{- 3}{5}

is incorrect
The following inequalities are correct:
(ii)

\frac{- 3}{5} < \frac{- 1}{5}

(iii)

\frac{3}{5} > \frac{1}{5}

(a) A Ball Is Dropped From A Height Of 30m. After Striking T...

Question 48. (a) Find wether the following numbers are equal or greater or less than

\frac{19}{26}

. The numbers are

\frac{247}{338}

\frac{19}{21}

and

\frac{26}{32}

.
(b) Find a rational number which lies between

\frac{1}{2}

and

\frac{1}{4}

.
[3 MARKS]

Discuss Question

:
(a) Answer: 2 Marks
(b) Answer: 1 Mark
(a) When we multiply by 13 in denominator and the numerator of

\frac{19}{26}

we get

\frac{247}{338}

, therefore

\frac{19}{26} = \frac{247}{338}

.
By inspection, when the numerator is same and the denominator is different then the fraction having a large value of denominator is smaller than the fraction having less value of denominator.
Hence,

\frac{19}{26}

\frac{19}{21}

.
For comparing

\frac{19}{26}

and

\frac{13}{16}

we have to make the denominator of both the fraction by taking LCM of 26 and 16 = 416.

\frac{(19 \times 16)}{(26 \times 16)} = \frac{304}{416}

\frac{(13 \times 26)}{(16 \times 26)} = \frac{338}{416}

338>304. Hence,

\frac{26}{32}

\frac{19}{26}

.
(b) Arational number which lies between

\frac{1}{2}

and

\frac{1}{4}

is:

\Rightarrow \frac{\frac{1}{2} + \frac{1}{4}}{2}

\Rightarrow \frac{\frac{3}{4}}{2}

\Rightarrow \frac{3}{8}

Question 49. What is the value of additive inverse of [2 MARKS]
(i)

\frac{7}{8}

(ii)

\frac{1}{2}

Discuss Question

:
Each part: 1 Mark
(i)Additive Inverse of

\frac{7}{8}

will be -

\frac{7}{8}

since the sum of both is 0.
(ii) Additive Inverse of

\frac{1}{2}

will be -

\frac{1}{2}

, since their sum will be 0.

Question 50. (a) Rahul's school is 10 km from his house. To cut the distance short he takes a shortcut. He first takes a route that is

\frac{4}{5}

of the original route but, he finds the road blocked at the end. The distance of his school from the blocked path is

\frac{1}{3}

of the original route. What was the difference in distance that Rahul covered by this new path and his original route?
(b) From a rope 68m long, pieces of equal size are cut. If the length of one piece is