APS, Golconda | Priyanka Gupta: CLASS 7 / worksheet on RATIONAL ... - Free Printable

Let's solve each problem step by step from the given worksheet.

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1. Write five rational numbers equivalent to $\frac{5}{-11}$

We can multiply both numerator and denominator by the same non-zero integer to get equivalent rational numbers.

$$
\frac{5}{-11} = \frac{-5}{11}
$$

Now, multiply numerator and denominator by 2, 3, 4, 5, 6:

- $ \frac{5 \times 2}{-11 \times 2} = \frac{10}{-22} $
- $ \frac{5 \times 3}{-11 \times 3} = \frac{15}{-33} $
- $ \frac{5 \times 4}{-11 \times 4} = \frac{20}{-44} $
- $ \frac{5 \times 5}{-11 \times 5} = \frac{25}{-55} $
- $ \frac{5 \times 6}{-11 \times 6} = \frac{30}{-66} $

Answer:
$$
\frac{10}{-22}, \frac{15}{-33}, \frac{20}{-44}, \frac{25}{-55}, \frac{30}{-66}
$$

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2. Express $\frac{9}{-15}$ as a rational number with:

First simplify $\frac{9}{-15} = -\frac{9}{15} = -\frac{3}{5}$

#### (i) Denominator 5

Already have $-\frac{3}{5}$ → so answer is:
$$
\boxed{-\frac{3}{5}}
$$

#### (ii) Numerator $-12$

We want $ \frac{-12}{?} = -\frac{3}{5} $

So:
$$
\frac{-12}{x} = -\frac{3}{5} \Rightarrow \frac{12}{x} = \frac{3}{5} \Rightarrow x = \frac{12 \times 5}{3} = 20
$$

So:
$$
\boxed{-\frac{12}{20}}
$$

#### (iii) Denominator 30

We want $ \frac{?}{30} = -\frac{3}{5} $

Multiply numerator and denominator by 6:
$$
-\frac{3 \times 6}{5 \times 6} = -\frac{18}{30}
$$

So:
$$
\boxed{-\frac{18}{30}}
$$

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3. Write each of the following in standard form:

Standard form means numerator and denominator are integers with no common factors, and denominator positive.

#### (i) $ \frac{78}{-91} $

Simplify: both divisible by 13.

- $78 ÷ 13 = 6$
- $91 ÷ 13 = 7$

So $ \frac{78}{-91} = -\frac{6}{7} $

Answer: $ \boxed{-\frac{6}{7}} $

#### (ii) $ \frac{-216}{162} $

Both divisible by 18:

- $216 ÷ 18 = 12$
- $162 ÷ 18 = 9$

So $ \frac{-216}{162} = -\frac{12}{9} = -\frac{4}{3} $

Answer: $ \boxed{-\frac{4}{3}} $

#### (iii) $ \frac{-195}{-520} $

Negative divided by negative = positive.

$ \frac{195}{520} $

Divide numerator and denominator by 5:

- $195 ÷ 5 = 39$
- $520 ÷ 5 = 104$

Now divide 39 and 104 by 13:

- $39 ÷ 13 = 3$
- $104 ÷ 13 = 8$

So $ \frac{3}{8} $

Answer: $ \boxed{\frac{3}{8}} $

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4. Which of the following are pairs of equivalent rational numbers?

Two rational numbers are equivalent if they reduce to the same fraction.

#### (i) $ \frac{-4}{13}, \frac{60}{-195} $

Simplify $ \frac{60}{-195} = -\frac{60}{195} $

Divide numerator and denominator by 15:

- $60 ÷ 15 = 4$
- $195 ÷ 15 = 13$

So $ -\frac{4}{13} $

Compare: $ \frac{-4}{13} $ vs $ -\frac{4}{13} $ → Same!

✔ Equivalent

#### (ii) $ \frac{7}{-15}, \frac{-35}{-75} $

Simplify first:

- $ \frac{7}{-15} = -\frac{7}{15} $
- $ \frac{-35}{-75} = \frac{35}{75} = \frac{7}{15} $

So: $ -\frac{7}{15} $ vs $ \frac{7}{15} $ → Not equal

✘ Not equivalent

#### (iii) $ \frac{16}{-20}, \frac{-56}{70} $

- $ \frac{16}{-20} = -\frac{16}{20} = -\frac{4}{5} $
- $ \frac{-56}{70} = -\frac{56}{70} = -\frac{4}{5} $

Same!

✔ Equivalent

Answer: (i) and (iii) are equivalent pairs.

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5. Which rational number is greater in each pair?

#### (i) $ \frac{-7}{9}, \frac{-5}{8} $

Both negative. Compare absolute values:

- $ \frac{7}{9} \approx 0.777 $
- $ \frac{5}{8} = 0.625 $

So $ \frac{7}{9} > \frac{5}{8} \Rightarrow -\frac{7}{9} < -\frac{5}{8} $

Thus, greater is $ \boxed{\frac{-5}{8}} $

#### (ii) $ \frac{-5}{8}, \frac{7}{-12} = -\frac{7}{12} $

Compare $ -\frac{5}{8} $ and $ -\frac{7}{12} $

Take LCM of 8 and 12 = 24

- $ -\frac{5}{8} = -\frac{15}{24} $
- $ -\frac{7}{12} = -\frac{14}{24} $

Since $ -\frac{14}{24} > -\frac{15}{24} $, so $ -\frac{7}{12} > -\frac{5}{8} $

Greater: $ \boxed{-\frac{7}{12}} $

#### (iii) $ \frac{11}{-16} = -\frac{11}{16}, \frac{-13}{18} $

Compare $ -\frac{11}{16} $ and $ -\frac{13}{18} $

LCM of 16 and 18 = 144

- $ -\frac{11}{16} = -\frac{99}{144} $
- $ -\frac{13}{18} = -\frac{104}{144} $

Now $ -\frac{99}{144} > -\frac{104}{144} $

So $ -\frac{11}{16} > -\frac{13}{18} $

Greater: $ \boxed{-\frac{11}{16}} $

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6. Arrange in ascending order

#### (i) $ \frac{-5}{6}, \frac{-17}{18}, \frac{23}{-24}, \frac{-11}{-13} $

Simplify all:

- $ \frac{-5}{6} $
- $ \frac{-17}{18} $
- $ \frac{23}{-24} = -\frac{23}{24} $
- $ \frac{-11}{-13} = \frac{11}{13} $ → positive

So only one positive: $ \frac{11}{13} $ — it will be the largest.

Now compare the negatives: $ -\frac{5}{6}, -\frac{17}{18}, -\frac{23}{24} $

Larger magnitude → more negative.

Compare fractions:

- $ \frac{5}{6} \approx 0.833 $
- $ \frac{17}{18} \approx 0.944 $
- $ \frac{23}{24} \approx 0.958 $

So $ -\frac{23}{24} < -\frac{17}{18} < -\frac{5}{6} $

Then $ \frac{11}{13} $ is largest.

So ascending order:

$$
\boxed{-\frac{23}{24}, -\frac{17}{18}, -\frac{5}{6}, \frac{11}{13}}
$$

#### (ii) $ \frac{-25}{6}, \frac{15}{-4}, \frac{-17}{8}, \frac{-53}{12} $

Simplify:

- $ -\frac{25}{6} \approx -4.166 $
- $ \frac{15}{-4} = -\frac{15}{4} = -3.75 $
- $ -\frac{17}{8} = -2.125 $
- $ -\frac{53}{12} \approx -4.416 $

Now order from smallest to largest:

- $ -\frac{53}{12} \approx -4.416 $
- $ -\frac{25}{6} \approx -4.166 $
- $ -\frac{15}{4} = -3.75 $
- $ -\frac{17}{8} = -2.125 $

So ascending order:

$$
\boxed{-\frac{53}{12}, -\frac{25}{6}, -\frac{15}{4}, -\frac{17}{8}}
$$

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7. Insert five rational numbers between $-\frac{3}{5}$ and $-\frac{1}{2}$

First convert to same denominator.

- $ -\frac{3}{5} = -\frac{6}{10} $
- $ -\frac{1}{2} = -\frac{5}{10} $

But $ -\frac{6}{10} < -\frac{5}{10} $, so we need numbers between $ -\frac{6}{10} $ and $ -\frac{5}{10} $

To find five numbers in between, use a larger denominator.

Let’s take denominator 100:

- $ -\frac{3}{5} = -\frac{60}{100} $
- $ -\frac{1}{2} = -\frac{50}{100} $

Now numbers between $ -\frac{60}{100} $ and $ -\frac{50}{100} $:

Choose: $ -\frac{59}{100}, -\frac{58}{100}, -\frac{57}{100}, -\frac{56}{100}, -\frac{55}{100} $

Simplify if needed:

- $ -\frac{59}{100} $
- $ -\frac{29}{50} $
- $ -\frac{57}{100} $
- $ -\frac{14}{25} $
- $ -\frac{11}{20} $

But any five distinct rational numbers in between work.

Answer (example):
$$
\boxed{-\frac{59}{100}, -\frac{58}{100}, -\frac{57}{100}, -\frac{56}{100}, -\frac{55}{100}}
$$
or simplified:
$$
\boxed{-\frac{59}{100}, -\frac{29}{50}, -\frac{57}{100}, -\frac{14}{25}, -\frac{11}{20}}
$$

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8. Find the sum:

#### (i) $ \frac{-2}{3} + \left( \frac{5}{-7} \right) = \frac{-2}{3} + \left( -\frac{5}{7} \right) $

LCM of 3 and 7 = 21

- $ \frac{-2}{3} = \frac{-14}{21} $
- $ -\frac{5}{7} = \frac{-15}{21} $

Sum: $ \frac{-14 - 15}{21} = \frac{-29}{21} $

Answer: $ \boxed{-\frac{29}{21}} $

#### (ii) $ -1\frac{1}{12} + \frac{-5}{9} $

Convert mixed number:

- $ -1\frac{1}{12} = -\frac{13}{12} $

Now: $ -\frac{13}{12} + (-\frac{5}{9}) = -\frac{13}{12} - \frac{5}{9} $

LCM of 12 and 9 = 36

- $ -\frac{13}{12} = -\frac{39}{36} $
- $ -\frac{5}{9} = -\frac{20}{36} $

Sum: $ \frac{-39 - 20}{36} = \frac{-59}{36} $

Answer: $ \boxed{-\frac{59}{36}} $

#### (iii) $ 2\frac{2}{5} + (-4\frac{3}{10}) $

Convert:

- $ 2\frac{2}{5} = \frac{12}{5} $
- $ -4\frac{3}{10} = -\frac{43}{10} $

Now: $ \frac{12}{5} - \frac{43}{10} $

LCM = 10

- $ \frac{12}{5} = \frac{24}{10} $
- $ \frac{24}{10} - \frac{43}{10} = \frac{-19}{10} = -1\frac{9}{10} $

Answer: $ \boxed{-\frac{19}{10}} $ or $ \boxed{-1\frac{9}{10}} $

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## ✔ Final Answers Summary:

1. Equivalent to $ \frac{5}{-11} $:
$ \frac{10}{-22}, \frac{15}{-33}, \frac{20}{-44}, \frac{25}{-55}, \frac{30}{-66} $

2. $ \frac{9}{-15} $:
(i) $ -\frac{3}{5} $
(ii) $ -\frac{12}{20} $
(iii) $ -\frac{18}{30} $

3. Standard form:
(i) $ -\frac{6}{7} $
(ii) $ -\frac{4}{3} $
(iii) $ \frac{3}{8} $

4. Equivalent pairs: (i) and (iii)

5. Greater:
(i) $ -\frac{5}{8} $
(ii) $ -\frac{7}{12} $
(iii) $ -\frac{11}{16} $

6. Ascending order:
(i) $ -\frac{23}{24}, -\frac{17}{18}, -\frac{5}{6}, \frac{11}{13} $
(ii) $ -\frac{53}{12}, -\frac{25}{6}, -\frac{15}{4}, -\frac{17}{8} $

7. Five rationals between $ -\frac{3}{5} $ and $ -\frac{1}{2} $:
Example: $ -\frac{59}{100}, -\frac{58}{100}, -\frac{57}{100}, -\frac{56}{100}, -\frac{55}{100} $

8. Sums:
(i) $ -\frac{29}{21} $
(ii) $ -\frac{59}{36} $
(iii) $ -\frac{19}{10} $ or $ -1\frac{9}{10} $

Let me know if you'd like these formatted neatly!