Fractions

Q1: A man earns ₹18720 per month and spends \(\frac{5}{6}\) of his income, Find his monthly

i. Find Monthly Expenditure:

Step 1: Given income = ₹18720
Fraction of income spent = \(\frac{5}{6}\)
Step 2: Calculate expenditure: \[ \text{Expenditure} = \frac{5}{6} \times 18720 \]Step 3: Multiply: \[ = \frac{5 \times 18720}{6} = \frac{93600}{6} = 15600 \] Answer: Monthly Expenditure = ₹15600

ii. Find Monthly Saving:

Step 4: Saving = Income – Expenditure \[ = 18720 – 15600 = 3120 \]Answer: Monthly Saving = ₹3120

Q2: A water tank can hold \(56\frac{1}{4}\) litres of water. How much water is contained in the tank when it is \(\frac{8}{15}\) full?

Step 1: Convert \(56\frac{1}{4}\) to improper fraction: \[ 56\frac{1}{4} = \frac{(56 \times 4) + 1}{4} = \frac{224 + 1}{4} = \frac{225}{4} \]Step 2: Multiply total capacity by fraction full: \[ \text{Water contained} = \frac{225}{4} \times \frac{8}{15} \]Step 3: Simplify before multiplying: \[ = \frac{225 \times 8}{4 \times 15} = \frac{225 \times 8}{60} \]Step 4: Simplify numerator and denominator: \[ 225 \div 15 = 15, \quad 60 \div 15 = 4 \] So, \[ = \frac{15 \times 8}{4} = \frac{120}{4} = 30 \]Answer: The water contained in the tank is 30 litres when it is \(\frac{8}{15}\) full.

Q3: After reading \(\frac{5}{8}\) of a book, 168 pages are left. How many pages are there in all in the book?

Step 1: If \(\frac{5}{8}\) of the book is read, then the remaining part of the book is: \[ 1 – \frac{5}{8} = \frac{8}{8} – \frac{5}{8} = \frac{3}{8} \]Step 2: Pages left after reading = \(\frac{3}{8}\) of total pages \(x\), which equals 168 pages: \[ \frac{3}{8} \times x = 168 \]Step 3: Multiply both sides by reciprocal of \(\frac{3}{8}\), which is \(\frac{8}{3}\): \[ x = 168 \times \frac{8}{3} \]Step 4: Simplify: \[ 168 \div 3 = 56 \] So, \[ x = 56 \times 8 = 448 \]Answer: The total number of pages in the book is 448 pages.

Q4: \(\frac{2}{3}\) of the students in a class are boys and the rest are 17 girls.

i. How many boys are there in the class?

Step 1: Let the total number of students in the class be \(x\).
Since \(\frac{2}{3}\) of the students are boys, the remaining fraction of girls is: \[ 1 – \frac{2}{3} = \frac{1}{3} \]Step 2: The number of girls is given as 17, so: \[ \frac{1}{3} \times x = 17 \]Step 3: Multiply both sides by 3 to find \(x\): \[ x = 17 \times 3 = 51 \]Answer: Number of boys = 34

ii. What is the total strength of the class?

Step 4: Total students \(= x = 51\).
Step 5: Number of boys \(= \frac{2}{3} \times 51 = 34\).
Answer: Total strength of the class = 51

Q5: A man earns ₹25440 per month. He spends \(\frac{1}{4}\) of it on house rent, \(\frac{3}{8}\) on food and clothes, \(\frac{1}{10}\) on insurance and \(\frac{1}{5}\) on other items. The rest he saves. How much does he save each month?

Step 1: Calculate the total fraction of income spent. \[ \frac{1}{4} + \frac{3}{8} + \frac{1}{10} + \frac{1}{5} \]Step 2: Find the LCM of denominators \(4, 8, 10, 5\): \[ \text{LCM} = 40 \]Express each fraction with denominator 40: \[ \frac{1}{4} = \frac{10}{40}, \quad \frac{3}{8} = \frac{15}{40}, \quad \frac{1}{10} = \frac{4}{40}, \quad \frac{1}{5} = \frac{8}{40} \]Step 3: Add the fractions: \[ \frac{10}{40} + \frac{15}{40} + \frac{4}{40} + \frac{8}{40} = \frac{37}{40} \]Step 4: Fraction saved = \[ 1 – \frac{37}{40} = \frac{3}{40} \]Step 5: Calculate the amount saved: \[ \frac{3}{40} \times 25440 = \frac{3 \times 25440}{40} = \frac{76320}{40} = 1908 \]Answer: The man saves ₹1908 each month.

Q6: An objective test was given to a group of 168 students. It was found that \(\frac{5}{6}\) of the students gave all correct answers. How many students made 1 or more mistakes?

Step 1: Find the number of students who gave all correct answers by calculating \(\frac{5}{6}\) of 168: \[ \frac{5}{6} \times 168 = \frac{5 \times 168}{6} \]Step 2: Simplify the multiplication: \[ = 5 \times 28 = 140 \]Step 3: Calculate the number of students who made 1 or more mistakes by subtracting the students who gave all correct answers from the total students: \[ 168 – 140 = 28 \]Answer: 28 students made 1 or more mistakes.

Q7: In an orchard, \(\frac{1}{3}\) of the trees are guava trees, \(\frac{1}{8}\) are banana trees and the rest are mango trees. If there are 117 mango trees in the orchard, how many trees in all are there?

Step 1: Add the fractions of guava and banana trees: \[ \frac{1}{3} + \frac{1}{8} = \frac{8}{24} + \frac{3}{24} = \frac{11}{24} \]Step 2: Find the fraction of mango trees by subtracting from 1: \[ 1 – \frac{11}{24} = \frac{24}{24} – \frac{11}{24} = \frac{13}{24} \]Step 3: Let the total number of trees be \(x\). Mango trees are \(\frac{13}{24}\) of total, which equals 117: \[ \frac{13}{24} \times x = 117 \]Step 4: Solve for \(x\): \[ x = \frac{117 \times 24}{13} \]Step 5: Calculate the value: \[ 117 \div 13 = 9, \quad \Rightarrow x = 9 \times 24 = 216 \]Answer: There are 216 trees in the orchard in all.

Q8: In a school, \(\frac{1}{25}\) of the students were absent on a certain day. If 720 students were present on that day, what is the total number of students in the school?

Step 1:
Total fraction of students = 1
Fraction absent = \(\frac{1}{25}\)
Fraction present = \[ 1 – \frac{1}{25} = \frac{25}{25} – \frac{1}{25} = \frac{24}{25} \]Step 2:
Let the total number of students be \(x\).
Number of students present = \(\frac{24}{25} \times x = 720\)
Step 3: Solve for \(x\): \[ x = \frac{720 \times 25}{24} \]Step 4: Calculate the value: \[ 720 \div 24 = 30 \\ x = 30 \times 25 = 750 \]Answer: The total number of students in the school is 750.

Q9: The product of three numbers is \(7\frac{1}{2}\). If two of them are \(1\frac{1}{7}\) and \(3\frac{3}{4}\), find the third number.

Step 1: \[ 7\frac{1}{2} = \frac{15}{2} \]Step 2: \[ 1\frac{1}{7} = \frac{8}{7} \]Step 3: \[ 3\frac{3}{4} = \frac{15}{4} \]Step 4: \[ \frac{15}{2} = \frac{8}{7} \times \frac{15}{4} \times x \]Step 5: Calculate the product of the two known numbers: \[ \frac{8}{7} \times \frac{15}{4} = \frac{8 \times 15}{7 \times 4} = \frac{120}{28} = \frac{30}{7} \]Step 6: Now, \[ \frac{15}{2} = \frac{30}{7} \times x \\ x = \frac{15}{2} \div \frac{30}{7} \]Step 7: Dividing fractions: \[ x = \frac{15}{2} \times \frac{7}{30} = \frac{15 \times 7}{2 \times 30} = \frac{105}{60} = \frac{7}{4} \]Answer: The third number is \(\frac{7}{4}\) or \(1\frac{3}{4}\).