GSEB Class 7 Maths Solutions Chapter 2 Fractions and Decimals InText Questions

Try These (Page 34)

Question 1. Find:
(a) \( \frac { 2 }{7} \times 3 \)
(b) \( \frac {9}{7} \times 6 \)
(c) \( 3 \times \frac {1}{8} \)
(d) \( \frac { 13 }{ 11 } \times 6 \)
If the product is an improper fraction express it as a mixed fraction.
Answer:
(a) \( \frac { 2 }{7} \times 3 = \frac {2 \times 3 }{7} = \frac { 6 }{7} \)
(b) \( \frac {9}{7} \times 6 = \frac { 54 }{7} = 7\frac {5}{7} \)
(c) \( 3 \times \frac {1}{8} = \frac {3 \times 1}{8} = \frac { 3 }{ 8 } \)
(d) \( \frac { 13 }{ 11 } \times 6 = \frac { 13 \times 6 }{ 11 } = \frac { 78 }{11} = 7\frac { 1 }{11} \)
In simple words: To find the product of a fraction and a whole number, you multiply the numerator of the fraction by the whole number. If the result is a top-heavy (improper) fraction, change it into a mixed number.

Exam Tip: Remember to simplify fractions where possible and convert any improper fractions to mixed numbers in their simplest form to get full marks.

Question 2. Represent pictorially: \( 2 \times \frac { 2 }{ 5 } = \frac { 4 }{ 5 } \)
Answer: The pictorial representation shows two groups, each with a circle divided into 5 equal parts, with 2 parts shaded. When combined, these shaded parts represent \( \frac{4}{5} \) of a whole circle.
For instance, imagine two circles, each split into five equal sections. In the first circle, two sections are colored in. In the second circle, two sections are also colored in. If you add up all the colored sections, you get four sections in total, which is like having four out of five parts of one circle shaded.

In simple words: When you have two groups of \( \frac{2}{5} \), you combine them to get \( \frac{4}{5} \). The image shows this by shading two parts in each of two circles, and then showing a single circle with four parts shaded.

Exam Tip: For pictorial representations, make sure your drawings clearly show the fractions and their sum. Use consistent shapes and divisions.

Try These (Page 34)

Question 1. Find:
(i) \( 5 \times 2\frac { 3 }{7} \)
(ii) \( 1\frac { 4 }{9} \times 6 \)
Answer:
(i) \( 5 \times 2\frac { 3 }{ 7 } = 5 \times \frac { (2 \times 7) + 3 }{7} = 5 \times \frac { 14+3 }{7} = 5 \times \frac { 17 }{7} = \frac { 5 \times 17 }{ 7 } = \frac { 85 }{ 7 } = 12\frac { 1 }{7} \)
(ii) \( 1\frac { 4 }{ 9 } \times 6 = \frac { (1 \times 9) + 4 }{9} \times 6 = \frac { 13 }{9} \times 6 = \frac { 13 \times 6 }{ 9 } = \frac { 78 }{9} = \frac { 26 }{ 3 } = 8\frac { 2 }{ 3 } \)
In simple words: First, change any mixed numbers into improper fractions. Then, multiply the numerators and keep the denominators the same. If the final answer is an improper fraction, convert it to a mixed number.

Exam Tip: Always remember to convert mixed fractions to improper fractions before performing multiplication. Simplify the fraction before converting back to a mixed number if possible.

Try These (Page 35)

Question 1. Can you tell, what is
(i) \( \frac { 1 }{ 2 } \) of 10?
(ii) \( \frac { 1 }{ 4 } \) of 16?
(iii) \( \frac { 2 }{ 5 } \) of 25?
Answer:
(i) \( \frac { 1 }{ 2 } \) of \( 10 = \frac { 1 }{ 2 } \times 10 = \frac { 1 \times 10 }{ 2 } = \frac { 10 }{ 2 } = 5 \)
(ii) \( \frac { 1 }{ 4 } \) of \( 16 = \frac { 1 }{ 4 } \times 16 = \frac { 1 \times 16 }{ 4 } = \frac { 16 }{ 4 } = 4 \)
(iii) \( \frac { 2 }{ 5 } \) of \( 25 = \frac { 2 }{ 5 } \times 25 = \frac { 2 \times 25 }{ 5 } = \frac { 50 }{ 5 } = 10 \)
In simple words: To find a fraction "of" a number, you multiply the fraction by that number. "Of" basically means "multiply" in math.

Exam Tip: Remember that "of" means multiplication in mathematics. Simplify the multiplication before performing the final division to prevent errors.

Try These (Page 39)

Question 1. Fill in these boxes:
(i) \( \frac { 1 }{ 2 } \times \frac {1}{7} = \frac { 1 \times 1 }{ 2 \times 7 } = \square \)
(ii) \( \frac { 1 }{ 5 } \times \frac { 1 }{ 7 } = \square \)
(iii) \( \frac { 1 }{ 7 } \times \frac { 1 }{ 2 } = \square \)
(iv) \( \frac { 1 }{ 7 } \times \frac { 1 }{ 5 } = \square \)
Answer:
(i) \( \frac { 1 }{ 2 } \times \frac {1}{7} = \frac { 1 \times 1 }{ 2 \times 7 } = \frac { 1 }{ 14 } \)
(ii) \( \frac { 1 }{ 5 } \times \frac { 1 }{ 7 } = \frac { 1 \times 1 }{ 5 \times 7 } = \frac { 1 }{ 35 } \)
(iii) \( \frac { 1 }{ 7 } \times \frac { 1 }{ 2 } = \frac { 1 \times 1 }{ 7 \times 2 } = \frac { 1 }{ 14 } \)
(iv) \( \frac { 1 }{ 7 } \times \frac { 1 }{ 5 } = \frac { 1 \times 1 }{ 7 \times 5 } = \frac { 1 }{ 35 } \)
In simple words: To multiply fractions, you simply multiply the top numbers together (numerators) and the bottom numbers together (denominators).

Exam Tip: Multiplying fractions is straightforward: numerator times numerator, denominator times denominator. No need for common denominators!

Try These (Page 40)

Question 1. Find:
(i) \( \frac {1}{ 3 } \times \frac { 4 }{ 5 } \)
(ii) \( \frac { 2 }{ 3 } \times \frac { 1 }{ 5 } \)
Answer:
(i) \( \frac { 1 }{ 3 } \times \frac { 4 }{ 5 } = \frac { 1 \times 4 }{ 3 \times 5 } = \frac { 4 }{ 15 } \)
(ii) \( \frac { 2 }{ 3 } \times \frac { 1 }{ 5 } = \frac { 2 \times 1 }{ 3 \times 5 } = \frac { 2 }{ 15 } \)
In simple words: When multiplying fractions, just multiply the numbers on top and then multiply the numbers on the bottom.

Exam Tip: Always check if you can cross-simplify before multiplying to make the numbers smaller and calculations easier.

Try These (Page 40)

Question 1. Find:
(i) \( \frac {8}{ 3 } \times \frac { 4 }{ 7 } \)
(ii) \( \frac { 3 }{ 4 } \times \frac { 2 }{ 3 } \)
Answer:
(i) \( \frac { 8 }{ 3 } \times \frac { 4 }{ 7 } = \frac { 8 \times 4 }{ 3 \times 7 } = \frac { 32 }{ 21 } \)
(ii) \( \frac { 3 }{ 4 } \times \frac { 2 }{ 3 } = \frac { 3 \times 2 }{ 4 \times 3 } = \frac { 6 }{ 12 } = \frac { 1 }{ 2 } \)
In simple words: Multiply the top numbers together and the bottom numbers together. Simplify your answer if you can by dividing both the top and bottom by the same number.

Exam Tip: Always look for common factors between numerators and denominators (even diagonally) to simplify the multiplication process and reduce the final fraction.

Think, Discuss and Write (Page 44)

Question 1.
(i) Will the reciprocal of a proper fraction be again a proper fraction?
(ii) Will the reciprocal of an improper fraction be again an improper fraction!
Answer:
(i) No, the reciprocal of a proper fraction is an improper fraction.
(ii) No, the reciprocal of an improper fraction is a proper fraction.
We can also state that:
(a) \( 1 \div \frac { 1 }{ 2 } = 1 \times \frac { 2 }{ 1 } = 1 \times \text{reciprocal of } \frac { 1 }{ 2 } \)
(b) \( 3 \div \frac { 1 }{ 4 } = 3 \times \frac { 4 }{1} = 3 \times \text{reciprocal of } \frac {1}{4} \)
(c) \( 3 \div \frac { 1 }{ 2 } = 3 \times \frac { 2 }{ 1 } = 3 \times \text{reciprocal of } \frac { 1 }{ 2 } \)
And, \( 2 \div \frac { 3 }{ 4 } = 2 \times \frac { 4 }{ 3 } \)
(d) \( 5 \div \frac { 2 }{ 9 } = 5 \times \frac { 9 }{ 2 } = 5 \times \text{reciprocal of } \frac { 2 }{ 9 } \)
I. When the product of two fractions is unity, then each is called the “reciprocal of the other".
II. When unity is divided by a fraction, then the quotient is the “reciprocal” of that fraction.
In simple words: A proper fraction is less than 1 (like \( \frac{1}{2} \)). Its reciprocal (like \( \frac{2}{1} \) or 2) will be greater than 1, making it an improper fraction or a whole number. An improper fraction is greater than 1 (like \( \frac{3}{2} \)). Its reciprocal (like \( \frac{2}{3} \)) will be less than 1, making it a proper fraction.

Exam Tip: Understand that proper fractions have a numerator smaller than the denominator, while improper fractions have a numerator larger than or equal to the denominator. Reciprocals flip these properties.

Try These (Page 45)

Question 1. Find:
(i) \( 7 \div \frac {2 }{ 5 } \)
(ii) \( 6 \div \frac { 4 }{ 7 } \)
(iii) \( 2 \div \frac { 8 }{9} \)
Answer:
(i) \( 7 \div \frac { 2 }{ 5 } = 7 \times \frac { 5 }{ 2 } = \frac { 7 \times 5 }{ 2 } = \frac { 35 }{ 2 } = 17\frac { 1 }{ 2 } \)
(ii) \( 6 \div \frac { 4 }{ 7 } = 6 \times \frac { 7 }{ 4 } = \frac { 6 \times 7 }{ 4 } = \frac { 42 }{ 4 } = \frac { 21 }{ 2 } = 10\frac { 1 }{ 2 } \)
(iii) \( 2 \div \frac { 8 }{ 9 } = 2 \times \frac { 9 }{ 8 } = \frac { 2 \times 9 }{ 8 } = \frac { 18 }{ 8 } = \frac { 9 }{ 4 } = 2\frac { 1 }{ 4 } \)
In simple words: To divide a whole number by a fraction, you flip the fraction (find its reciprocal) and then multiply it by the whole number.

Exam Tip: Remember the "Keep, Change, Flip" method for dividing fractions: Keep the first number, Change division to multiplication, Flip the second fraction (use its reciprocal).

Try These (Page 45)

Question 1. Find:
(i) \( 6 + 5\frac { 1 }{ 3 } \)
(ii) \( 7 + 2\frac { 4 }{ 7 } \)
Answer:
(i) \( 6 + 5\frac { 1 }{ 3 } = 6 + \frac { 16 }{ 3 } = \frac { 6 \times 3 }{ 3 } + \frac { 16 }{ 3 } = \frac { 18 }{ 3 } + \frac { 16 }{ 3 } = \frac { 18 + 16 }{ 3 } = \frac { 34 }{ 3 } = 11\frac{1}{3} \)
(ii) \( 7 + 2\frac { 4 }{ 7 } = 7 + \frac { 18 }{ 7 } = \frac { 7 \times 7 }{ 7 } + \frac { 18 }{ 7 } = \frac { 49 }{ 7 } + \frac { 18 }{ 7 } = \frac { 49 + 18 }{ 7 } = \frac { 67 }{ 7 } = 9\frac { 4 }{ 7 } \)
In simple words: To add a whole number and a mixed number, you can change the mixed number into an improper fraction. Then, find a common bottom number (denominator) for both, add the top numbers, and convert the result back to a mixed number if needed.

Exam Tip: When adding whole numbers and fractions, always convert the whole number into a fraction with the same denominator as the other fraction or convert the mixed number to an improper fraction for easier calculation.

Try These (Page 45)

Question 1. Find:
(i) \( \frac { 3 }{ 5 } \div \frac { 1 }{ 2 } \)
(ii) \( \frac {1}{ 2 } \div \frac { 3 }{5} \)
(iii) \( 2\frac { 1 }{ 2 } \div \frac { 3 }{ 5 } \)
(iv) \( 5\frac { 1 }{6} \div \frac { 9 }{ 2 } \)
Answer:
(i) \( \frac { 3 }{ 5 } \div \frac { 1 }{ 2 } = \frac { 3 }{ 5 } \times \frac { 2 }{ 1 } = \frac { 3 \times 2 }{ 5 \times 1 } = \frac { 6 }{ 5 } = 1\frac { 1 }{ 5 } \)
(ii) \( \frac { 1 }{ 2 } \div \frac { 3 }{ 5 } = \frac { 1 }{ 2 } \times \frac { 5 }{ 3 } = \frac { 1 \times 5 }{ 2 \times 3 } = \frac { 5 }{ 6 } \)
(iii) \( 2\frac { 1 }{ 2 } \div \frac { 3 }{ 5 } = \frac { 5 }{ 2 } \div \frac { 3 }{ 5 } = \frac { 5 }{ 2 } \times \frac { 5 }{ 3 } = \frac { 5 \times 5 }{ 2 \times 3 } = \frac { 25 }{ 6 } = 4\frac { 1 }{ 6 } \)
(iv) \( 5\frac { 1 }{6} \div \frac { 9 }{ 2 } = \frac { 31 }{ 6 } \div \frac { 9 }{ 2 } = \frac { 31 }{ 6 } \times \frac { 2 }{ 9 } = \frac { 31 \times 2 }{ 6 \times 9 } = \frac { 62 }{ 54 } = \frac { 31 }{ 27 } = 1\frac { 4 }{ 27 } \)
In simple words: To divide fractions, you need to turn the second fraction upside down (find its reciprocal) and then multiply it by the first fraction. If there's a mixed number, change it to an improper fraction first.

Exam Tip: Always convert mixed numbers to improper fractions before performing division. Remember to simplify the resulting fraction if possible.

Try These (Page 50)

Question 1. Find:
(i) \( 2.7 \times 4 \)
(ii) \( 1.8 \times 1.2 \)
(iii) \( 2.3 \times 4.35 \)
Answer:
(i) \( 2.7 \times 4 \)
Multiply \( 27 \times 4 = 108 \). Since there is one digit to the right of the decimal point in 2.7, the result also has one digit to the right of the decimal point.
\( \implies 2.7 \times 4 = 10.8 \)
(ii) \( 1.8 \times 1.2 \)
Multiply \( 18 \times 12 = 216 \). There is one digit after the decimal in 1.8 and one digit after the decimal in 1.2, so there are \( (1+1) = 2 \) digits after the decimal point in the final product.
\( \implies 1.8 \times 1.2 = 2.16 \)
(iii) \( 2.3 \times 4.35 \)
Multiply \( 23 \times 435 = 10005 \). There is one digit after the decimal in 2.3 and two digits after the decimal in 4.35, so there are \( (1+2) = 3 \) digits after the decimal point in the final product.
\( \implies 2.3 \times 4.35 = 10.005 \)
In simple words: To multiply decimal numbers, first ignore the decimal points and multiply the numbers like whole numbers. Then, count the total number of digits after the decimal point in all the numbers you multiplied. Your answer will have that same total number of digits after its decimal point.

Exam Tip: The number of decimal places in the product is the sum of the decimal places in the numbers being multiplied. This rule is crucial for accuracy.

Question 2. Arrange the products obtained in Question 1 in descending order.
Answer: The products are: 10.8, 2.16, 10.005.
Comparing 10.8 and 10.005, we observe:
\( 10 = 10 \), and \( 8 > 0 \), meaning \( 10.005 < 10.8 \).
Here, the smallest number is 2.16, and the largest number is 10.8.
Thus, the required descending order is: 10.8, 10.005, 2.16.
In simple words: We need to list the answers from the previous question from the biggest to the smallest. Comparing 10.8, 2.16, and 10.005, the order from largest to smallest is 10.8, then 10.005, and finally 2.16.

Exam Tip: To compare decimals, always align the decimal points and compare digits from left to right, starting with the largest place value. If one number has more digits after the decimal, you can add trailing zeros to the other numbers to make them equal in length for easier comparison.

Try These (Page 51)

Question 1. Find:
(i) \( 0.3 \times 10 \)
(ii) \( 1.2 \times 100 \)
(iii) \( 56.3 \times 1000 \)
Answer:
(i) \( 0.3 \times 10 \)
There is 1 zero in 10. So, the decimal point is shifted to the right by 1 place.
Thus, \( 0.3 \times 10 = 3 \)
(ii) \( 1.2 \times 100 \)
There are 2 zeros in 100. So, the decimal point is shifted to the right by 2 places.
Thus, \( 1.2 \times 100 = 120 \)
(iii) \( 56.3 \times 1000 \)
There are three zeros in 1000. So, the decimal point is shifted to the right by 3 places.
Thus, \( 56.3 \times 1000 = 56300 \)
In simple words: When you multiply a decimal number by 10, 100, or 1000, you move the decimal point to the right. The number of places you move it is equal to the number of zeros in 10, 100, or 1000.

Exam Tip: Multiplying by powers of 10 (10, 100, 1000, etc.) moves the decimal point to the right. The number of places moved corresponds to the number of zeros in the power of 10.

Try These (Page 53)

Question 1. Find:
(i) \( 235.4 \div 10 \)
(ii) \( 235.4 \div 100 \)
(iii) \( 235.4 \div 1000 \)
Answer:
(i) \( 235.4 \div 10 \)
Since there is one zero in 10, the decimal point in the quotient is shifted to the left by one place.
\( \implies 235.4 \div 10 = 23.54 \)
(ii) \( 235.4 \div 100 \)
Since there are two zeros in 100, the decimal point in the quotient is shifted to the left by two places.
\( \implies 235.4 \div 100 = 2.354 \)
(iii) \( 235.4 \div 1000 \)
Since there are three zeros in 1000, the decimal point in the quotient is shifted to the left by three places.
\( \implies 235.4 \div 1000 = 0.2354 \)
In simple words: When you divide a decimal number by 10, 100, or 1000, you move the decimal point to the left. The number of places you move it is the same as the number of zeros in the divisor (10, 100, or 1000).

Exam Tip: Dividing by powers of 10 (10, 100, 1000, etc.) moves the decimal point to the left. The number of places moved matches the count of zeros in the power of 10.

Try These (Page 53)

Question 1. Find:
(i) \( 35.7 \div 3 \)
(ii) \( 25.5 \div 3 \)
Answer:
(i) \( 35.7 \div 3 \)
We know that \( 357 \div 3 = 119 \). Since there is one digit in the decimal part of 35.7, the decimal point is placed in the quotient after one digit from the rightmost digit.
\( \implies 35.7 \div 3 = 11.9 \)
(ii) \( 25.5 \div 3 \)
We know that \( 255 \div 3 = 85 \). Since there is one digit in the decimal part of 25.5, the decimal point is placed in the quotient after one digit from the rightmost digit.
\( \implies 25.5 \div 3 = 8.5 \)
In simple words: To divide a decimal by a whole number, you first divide the numbers like normal whole numbers. Then, you put the decimal point in the answer directly above the decimal point in the number you are dividing.

Exam Tip: When dividing a decimal by a whole number, ensure the decimal point in the quotient is directly above the decimal point in the dividend. Treat the division as if there were no decimal first, then place it correctly.

Try These (Page 53)

Question 1. Find:
(i) \( 43.15 \div 5 \)
(ii) \( 82.44 \div 6 \)
Answer:
(i) \( 43.15 \div 5 \)
Since \( 4315 \div 5 = 863 \) and there are two digits in the decimal part of 43.15.
So, place the decimal point in 863 such that there are two digits to its right.
\( \implies 43.15 \div 5 = 8.63 \)
(ii) \( 82.44 \div 6 \)
Since \( 8244 \div 6 = 1374 \) and there are two digits in the decimal part of 82.44.
So, place the decimal point in 1374 such that there are two digits to its right.
\( \implies 82.44 \div 6 = 13.74 \)
In simple words: When dividing a decimal by a whole number, perform the division as if they were whole numbers. Then, count the number of decimal places in the number being divided, and put that same number of decimal places in your answer.

Exam Tip: Count the number of decimal places in the dividend; this number determines the position of the decimal point in the quotient, ensuring correct placement.

Try These (Page 53)

Question 1. Find:
(i) \( 15.5 \div 5 \)
(ii) \( 126.35 \div 7 \)
Answer:
(i) \( 15.5 \div 5 \)
Since \( 155 \div 5 = 31 \) and there is one digit in the decimal part of 15.5.
So, place the decimal point in 31 such that there is one digit to its right.
\( \implies 15.5 \div 5 = 3.1 \)
(ii) \( 126.35 \div 7 \)
Since \( 12635 \div 7 = 1805 \) and there are two digits in the decimal part of 126.35.
So, place the decimal point in 1805 such that there are two digits to its right.
\( \implies 126.35 \div 7 = 18.05 \)
In simple words: First, divide the numbers without considering the decimal. Then, count how many numbers are after the decimal in the original problem. That's how many numbers should be after the decimal in your answer.

Exam Tip: The position of the decimal point in the quotient is directly determined by the number of decimal places in the dividend. Always ensure this correspondence for accurate results.

Try These (Page 54)

Question 1. Find:
(i) \( \frac { 7.75 }{ 0.25 } \)
(ii) \( \frac { 42.8 }{ 0.02 } \)
(iii) \( \frac { 5.6 }{ 1.4 } \)
Answer:
(i) \( \frac { 7.75 }{ 0.25 } \)
Since, \( 7.75 = \frac { 775 }{ 100 } \) and \( 0.25 = \frac {25}{ 100 } \)
\( \implies 7.75 \div 0.25 = \frac { 775 }{ 100 } \div \frac { 25 }{ 100 } \)
\( = \frac { 775 }{ 100 } \times \frac { 100 }{ 25 } \)
\( = \frac { 775 }{ 25 } = 31 \)
\( \implies \frac { 7.75 }{ 0.25 } = 31 \)
(ii) \( \frac { 42.8 }{ 0.02 } \)
Since, \( 42.8 = \frac { 428 }{ 10 } \) and \( 0.02 = \frac { 2 }{ 100 } \)
\( \implies 42.8 \div 0.02 = \frac { 428 }{ 10 } \div \frac { 2 }{ 100 } \)
\( = \frac { 428 }{ 10 } \times \frac { 100 }{ 2 } \)
\( = \frac { 428 \times 10 }{ 2 } = 214 \times 10 = 2140 \)
\( \implies \frac { 42.8 }{ 0.02 } = 2140 \)
(iii) \( \frac { 5.6 }{ 1.4 } \)
Since, \( 5.6 = \frac { 56 }{10} \) and \( 1.4 = \frac { 14 }{ 10 } \)
\( \implies 5.6 \div 1.4 = \frac { 56 }{ 10 } \div \frac { 14 }{ 10 } \)
\( = \frac { 56 }{ 10 } \times \frac { 10 }{ 14 } \)
\( = \frac { 56 }{ 14 } = 4 \)
\( \implies \frac { 5.6 }{ 1.4 } = 4 \)
In simple words: To divide a decimal by another decimal, first change both numbers into fractions without decimals by multiplying them by powers of 10. Then, divide the fractions by multiplying the first by the reciprocal of the second, and simplify.

Exam Tip: When dividing by a decimal, always convert the divisor to a whole number by multiplying both the divisor and dividend by the same power of 10. This simplifies the division process.