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Remedial Worksheet – Grade 7

#### Rational Numbers

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1. List three rational numbers between:

(a) -2 and -1
To find rational numbers between -2 and -1, we can choose any fractions or decimals that lie strictly between these two numbers. For example:
- $-\frac{5}{3} = -1.\overline{6}$
- $-\frac{4}{3} = -1.\overline{3}$
- $-\frac{3}{2} = -1.5$

So, the three rational numbers are:
$$
\boxed{-\frac{5}{3}, -\frac{4}{3}, -\frac{3}{2}}
$$

(b) $-\frac{4}{5}$ and $-\frac{2}{3}$
First, convert both fractions to a common denominator to compare them more easily:
- $-\frac{4}{5} = -\frac{12}{15}$
- $-\frac{2}{3} = -\frac{10}{15}$

Now, we need rational numbers between $-\frac{12}{15}$ and $-\frac{10}{15}$. For example:
- $-\frac{11}{15}$
- $-\frac{23}{30} = -\frac{23 \div 1}{30 \div 1} = -\frac{23}{30}$
- $-\frac{47}{60} = -\frac{47 \div 1}{60 \div 1} = -\frac{47}{60}$

So, the three rational numbers are:
$$
\boxed{-\frac{11}{15}, -\frac{23}{30}, -\frac{47}{60}}
$$

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2. Give four rational numbers equivalent to $-\frac{2}{7}$

Equivalent fractions are obtained by multiplying both the numerator and the denominator by the same non-zero integer. For example:
- $-\frac{2}{7} \times \frac{2}{2} = -\frac{4}{14}$
- $-\frac{2}{7} \times \frac{3}{3} = -\frac{6}{21}$
- $-\frac{2}{7} \times \frac{4}{4} = -\frac{8}{28}$
- $-\frac{2}{7} \times \frac{5}{5} = -\frac{10}{35}$

So, the four equivalent rational numbers are:
$$
\boxed{-\frac{4}{14}, -\frac{6}{21}, -\frac{8}{28}, -\frac{10}{35}}
$$

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3. Write the following rational numbers in simplest form:

(a) $-\frac{44}{72}$
Simplify by finding the greatest common divisor (GCD) of 44 and 72, which is 4:
$$
-\frac{44 \div 4}{72 \div 4} = -\frac{11}{18}
$$
So, the simplest form is:
$$
\boxed{-\frac{11}{18}}
$$

(b) $\frac{80}{-120}$
Simplify by finding the GCD of 80 and 120, which is 40:
$$
\frac{80 \div 40}{-120 \div 40} = \frac{2}{-3} = -\frac{2}{3}
$$
So, the simplest form is:
$$
\boxed{-\frac{2}{3}}
$$

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4. Fill in the boxes with the correct symbol ($<$, $>$, $=$):

(a) $-\frac{7}{21} \square -\frac{3}{9}$
Simplify both fractions:
- $-\frac{7}{21} = -\frac{1}{3}$
- $-\frac{3}{9} = -\frac{1}{3}$

Since both fractions are equal:
$$
-\frac{7}{21} = -\frac{3}{9}
$$
So, the symbol is:
$$
\boxed{=}
$$

(b) $-\frac{2}{3} \square \frac{2}{3}$
Compare the signs:
- $-\frac{2}{3}$ is negative.
- $\frac{2}{3}$ is positive.

Since any negative number is less than any positive number:
$$
-\frac{2}{3} < \frac{2}{3}
$$
So, the symbol is:
$$
\boxed{<}
$$

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5. Write the additive inverse of the following Rational numbers:

(a) $-\frac{8}{5}$
The additive inverse of a number $x$ is $-x$. So, the additive inverse of $-\frac{8}{5}$ is:
$$
-\left(-\frac{8}{5}\right) = \frac{8}{5}
$$
So, the additive inverse is:
$$
\boxed{\frac{8}{5}}
$$

(b) $-\frac{6}{7}$
The additive inverse of $-\frac{6}{7}$ is:
$$
-\left(-\frac{6}{7}\right) = \frac{6}{7}
$$
So, the additive inverse is:
$$
\boxed{\frac{6}{7}}
$$

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6. Add:

(a) $-\frac{8}{9} + \frac{-3}{5}$
Find a common denominator for 9 and 5, which is 45:
$$
-\frac{8}{9} = -\frac{8 \times 5}{9 \times 5} = -\frac{40}{45}
$$
$$
\frac{-3}{5} = -\frac{3 \times 9}{5 \times 9} = -\frac{27}{45}
$$
Now add the fractions:
$$
-\frac{40}{45} + \left(-\frac{27}{45}\right) = -\frac{40 + 27}{45} = -\frac{67}{45}
$$
So, the sum is:
$$
\boxed{-\frac{67}{45}}
$$

(b) $\frac{11}{15} + \frac{3}{-5}$
Find a common denominator for 15 and 5, which is 15:
$$
\frac{11}{15} = \frac{11}{15}
$$
$$
\frac{3}{-5} = -\frac{3 \times 3}{5 \times 3} = -\frac{9}{15}
$$
Now add the fractions:
$$
\frac{11}{15} + \left(-\frac{9}{15}\right) = \frac{11 - 9}{15} = \frac{2}{15}
$$
So, the sum is:
$$
\boxed{\frac{2}{15}}
$$

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7. Subtract $\frac{1}{7}$ from $\frac{-3}{8}$

Subtracting $\frac{1}{7}$ from $\frac{-3}{8}$ means:
$$
\frac{-3}{8} - \frac{1}{7}
$$
Find a common denominator for 8 and 7, which is 56:
$$
\frac{-3}{8} = -\frac{3 \times 7}{8 \times 7} = -\frac{21}{56}
$$
$$
\frac{1}{7} = \frac{1 \times 8}{7 \times 8} = \frac{8}{56}
$$
Now subtract the fractions:
$$
-\frac{21}{56} - \frac{8}{56} = -\frac{21 + 8}{56} = -\frac{29}{56}
$$
So, the result is:
$$
\boxed{-\frac{29}{56}}
$$

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8. Find the product of the reciprocal of $\frac{5}{7}$ and $\frac{9}{11}$

The reciprocal of $\frac{5}{7}$ is $\frac{7}{5}$. Now multiply:
$$
\frac{7}{5} \times \frac{9}{11} = \frac{7 \times 9}{5 \times 11} = \frac{63}{55}
$$
So, the product is:
$$
\boxed{\frac{63}{55}}
$$

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9. Product of two rational numbers is $\frac{27}{56}$. If one of the rational numbers is $-\frac{3}{7}$, find the other.

Let the other rational number be $x$. Then:
$$
-\frac{3}{7} \times x = \frac{27}{56}
$$
Solve for $x$:
$$
x = \frac{27}{56} \div \left(-\frac{3}{7}\right) = \frac{27}{56} \times \left(-\frac{7}{3}\right) = \frac{27 \times (-7)}{56 \times 3} = \frac{-189}{168}
$$
Simplify $\frac{-189}{168}$ by finding the GCD of 189 and 168, which is 21:
$$
\frac{-189 \div 21}{168 \div 21} = \frac{-9}{8}
$$
So, the other rational number is:
$$
\boxed{-\frac{9}{8}}
$$

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10. Ankit reads $\frac{1}{3}$ of a story book on the first day and $\frac{1}{4}$ of the book on the second day. What part of the book is yet to be read by Ankit?

Total part of the book read:
$$
\frac{1}{3} + \frac{1}{4}
$$
Find a common denominator for 3 and 4, which is 12:
$$
\frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}
$$
$$
\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}
$$
Now add the fractions:
$$
\frac{4}{12} + \frac{3}{12} = \frac{7}{12}
$$
Part of the book yet to be read:
$$
1 - \frac{7}{12} = \frac{12}{12} - \frac{7}{12} = \frac{5}{12}
$$
So, the part of the book yet to be read is:
$$
\boxed{\frac{5}{12}}
$$

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Comparing Quantities

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1. Express the following as ratios in the simplest form:

(a) 2 hours to 300 minutes
Convert 2 hours to minutes:
$$
2 \text{ hours} = 2 \times 60 = 120 \text{ minutes}
$$
Now express the ratio:
$$
120 \text{ minutes} : 300 \text{ minutes} = \frac{120}{300}
$$
Simplify by dividing both terms by their GCD, which is 60:
$$
\frac{120 \div 60}{300 \div 60} = \frac{2}{5}
$$
So, the ratio is:
$$
\boxed{2:5}
$$

(b) 5 km to 1 km 500 metres
Convert everything to the same unit (metres):
- 5 km = $5 \times 1000 = 5000$ metres
- 1 km 500 metres = $1 \times 1000 + 500 = 1500$ metres
Now express the ratio:
$$
5000 \text{ metres} : 1500 \text{ metres} = \frac{5000}{1500}
$$
Simplify by dividing both terms by their GCD, which is 500:
$$
\frac{5000 \div 500}{1500 \div 500} = \frac{10}{3}
$$
So, the ratio is:
$$
\boxed{10:3}
$$

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2. Convert the given decimals to percent:

(a) 0.65
To convert a decimal to a percentage, multiply by 100:
$$
0.65 \times 100 = 65\%
$$
So, the percentage is:
$$
\boxed{65\%}
$$

(b) 0.07
Similarly:
$$
0.07 \times 100 = 7\%
$$
So, the percentage is:
$$
\boxed{7\%}
$$

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3. Convert the following fractions to percentage:

(a) $\frac{2}{5}$
Convert to a decimal:
$$
\frac{2}{5} = 0.4
$$
Now convert to a percentage:
$$
0.4 \times 100 = 40\%
$$
So, the percentage is:
$$
\boxed{40\%}
$$

(b) $\frac{4}{20}$
Simplify the fraction:
$$
\frac{4}{20} = \frac{1}{5}
$$
Convert to a decimal:
$$
\frac{1}{5} = 0.2
$$
Now convert to a percentage:
$$
0.2 \times 100 = 20\%
$$
So, the percentage is:
$$
\boxed{20\%}
$$

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4. Find:

(a) 25% of 40
Convert 25% to a decimal:
$$
25\% = 0.25
$$
Now calculate:
$$
0.25 \times 40 = 10
$$
So, 25% of 40 is:
$$
\boxed{10}
$$

(b) 10% of 250
Convert 10% to a decimal:
$$
10\% = 0.1
$$
Now calculate:
$$
0.1 \times 250 = 25
$$
So, 10% of 250 is:
$$
\boxed{25}
$$

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Final Answer:

$$
\boxed{\text{See detailed solutions above.}}
$$