In a school 3/8 of the number of students are girls and the rest are boys. One-third of the number of boys are below 10 years and 2/3 the number if girls are also below 10 years. If the number of students of age 10 or more years is 260. then the number of boys in the school is:
300
Understanding the School Student Distribution Problem
The problem describes a school with a certain number of students, divided into boys and girls. We are given the fraction of students who are girls and, consequently, the fraction who are boys. The students are also categorized by age: those below 10 years and those aged 10 or more years. We are given the fraction of boys below 10 and the fraction of girls below 10. Finally, we know the total number of students aged 10 or more, and we need to find the total number of boys in the school.
Let's break down the information given:
- Fraction of students who are girls = $\frac{3}{8}$
- Fraction of students who are boys = $1 - \text{Fraction of girls}$
- Fraction of boys below 10 years = $\frac{1}{3}$
- Fraction of girls below 10 years = $\frac{2}{3}$
- Number of students aged 10 or more years = 260
Calculating Fractions of Students by Gender and Age
Let the total number of students in the school be $T$.
Based on the given fractions, we can express the number of boys and girls in terms of $T$:
- Number of girls = $\frac{3}{8} T$
- Number of boys = $(1 - \frac{3}{8}) T = \frac{5}{8} T$
Now, let's find the fraction of students in different age groups for both boys and girls:
- Fraction of boys below 10 years = $\frac{1}{3}$ of boys = $\frac{1}{3} \times \frac{5}{8} T = \frac{5}{24} T$
- Fraction of boys aged 10 or more years = (Total boys) - (Boys below 10) = $\frac{5}{8} T - \frac{5}{24} T$
To subtract these fractions, we find a common denominator, which is 24.
$\frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24}$
So, Fraction of boys aged 10 or more years = $\frac{15}{24} T - \frac{5}{24} T = \frac{10}{24} T = \frac{5}{12} T$
- Fraction of girls below 10 years = $\frac{2}{3}$ of girls = $\frac{2}{3} \times \frac{3}{8} T = \frac{6}{24} T = \frac{1}{4} T$
- Fraction of girls aged 10 or more years = (Total girls) - (Girls below 10) = $\frac{3}{8} T - \frac{1}{4} T$
To subtract these fractions, we find a common denominator, which is 8.
$\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}$
So, Fraction of girls aged 10 or more years = $\frac{3}{8} T - \frac{2}{8} T = \frac{1}{8} T$
Finding the Total Number of Students
We know that the total number of students aged 10 or more years is 260. This total is the sum of boys aged 10 or more and girls aged 10 or more.
Number of students aged 10 or more = (Fraction of boys aged 10 or more) + (Fraction of girls aged 10 or more)
$260 = \frac{5}{12} T + \frac{1}{8} T$
To add the fractions on the right side, we find a common denominator for 12 and 8, which is 24.
- $\frac{5}{12} = \frac{5 \times 2}{12 \times 2} = \frac{10}{24}$
- $\frac{1}{8} = \frac{1 \times 3}{8 \times 3} = \frac{3}{24}$
So, $260 = \frac{10}{24} T + \frac{3}{24} T = \frac{13}{24} T$
Now, we can solve for $T$:
$T = 260 \times \frac{24}{13}$
Since $260 \div 13 = 20$, we have:
$T = 20 \times 24 = 480$
The total number of students in the school is 480.
Calculating the Number of Boys
The problem asks for the number of boys in the school. We found earlier that the number of boys is $\frac{5}{8}$ of the total number of students ($T$).
Number of boys = $\frac{5}{8} T$
Substitute the value of $T = 480$:
Number of boys = $\frac{5}{8} \times 480$
Since $480 \div 8 = 60$, we have:
Number of boys = $5 \times 60 = 300$
Thus, the number of boys in the school is 300.
| Category | Fraction of Total Students | Number of Students (if T=480) |
|---|---|---|
| Total Students | $T$ | 480 |
| Girls | $\frac{3}{8} T$ | $\frac{3}{8} \times 480 = 180$ |
| Boys | $\frac{5}{8} T$ | $\frac{5}{8} \times 480 = 300$ |
| Boys below 10 | $\frac{5}{24} T$ | $\frac{5}{24} \times 480 = 100$ |
| Boys ≥ 10 | $\frac{5}{12} T$ | $\frac{5}{12} \times 480 = 200$ |
| Girls below 10 | $\frac{1}{4} T$ | $\frac{1}{4} \times 480 = 120$ |
| Girls ≥ 10 | $\frac{1}{8} T$ | $\frac{1}{8} \times 480 = 60$ |
| Total ≥ 10 | $\frac{13}{24} T$ | $200 + 60 = 260$ (Matches given information) |
Revision Table: Key Steps to Solve Student Distribution Problems
| Step | Description | Applied in this Problem |
|---|---|---|
| 1 | Identify total unknown quantity (e.g., Total Students). Represent it with a variable (e.g., $T$). | Let Total Students = $T$. |
| 2 | Express different categories (e.g., boys/girls) as fractions of the total. | Girls = $\frac{3}{8} T$, Boys = $\frac{5}{8} T$. |
| 3 | Express sub-categories (e.g., age groups within boys/girls) as fractions of the total. | Boys < 10 = $\frac{1}{3} \times \frac{5}{8} T = \frac{5}{24} T$, Boys ≥ 10 = $\frac{5}{8} T - \frac{5}{24} T = \frac{5}{12} T$. Girls < 10 = $\frac{2}{3} \times \frac{3}{8} T = \frac{1}{4} T$, Girls ≥ 10 = $\frac{3}{8} T - \frac{1}{4} T = \frac{1}{8} T$. |
| 4 | Use the given numerical information to form an equation involving the variable $T$. | Students ≥ 10 = (Boys ≥ 10) + (Girls ≥ 10). $260 = \frac{5}{12} T + \frac{1}{8} T = \frac{13}{24} T$. |
| 5 | Solve the equation to find the value of the total quantity $T$. | $\frac{13}{24} T = 260 \implies T = 260 \times \frac{24}{13} = 480$. |
| 6 | Use the value of $T$ to find the specific number asked for in the question. | Number of boys = $\frac{5}{8} T = \frac{5}{8} \times 480 = 300$. |
Additional Information: Working with Fractions in Word Problems
Word problems involving fractions require careful reading to understand what each fraction refers to (e.g., fraction of the total, or fraction of a subgroup). Here are some key points:
- Identify the Whole: Determine what quantity the fractions are based on. In this problem, some fractions ($\frac{3}{8}$) are based on the total number of students, while others ($\frac{1}{3}$, $\frac{2}{3}$) are based on subgroups (boys or girls).
- Calculate Remaining Fractions: If a fraction of a whole is given, the remaining part is $1 - \text{fraction}$. For example, if $\frac{3}{8}$ are girls, $1 - \frac{3}{8} = \frac{5}{8}$ are boys.
- Fraction of a Fraction: When you have a fraction of a subgroup (which is itself a fraction of the total), multiply the fractions to find the fraction relative to the total. For example, $\frac{1}{3}$ of the boys ($\frac{5}{8}$ of total) is $\frac{1}{3} \times \frac{5}{8} = \frac{5}{24}$ of the total.
- Combining Fractions: To find the total of different subgroups that meet a certain criterion (like age 10 or more), add their respective fractions relative to the total. Remember to use a common denominator for addition and subtraction of fractions.
- Setting up Equations: Use the given numerical value (like 260 students) to set up an equation where the sum of relevant fractions of the total quantity ($T$) equals the given number.
- Solving for the Variable: Once the equation is set up, solve for the unknown total ($T$) by isolating the variable. This usually involves multiplying by the reciprocal of the fraction multiplying $T$.
Practice with different types of fraction problems helps in quickly identifying how the fractions relate to the overall total and setting up the correct calculations.
Important Questions from Ratio and Proportion
The cost of a diamond is directly proportional to the square of its weight. The cost of a 14 gm diamond is Rs. 2560. This diamond got broken down into two pieces in the ratio of 5 ∶ 9. How much loss percent is incurred due to this breakage ? (Correct to two decimal places)
Atul purchased Bread costing Rs.20 and gave a 100 rupee note to the shopkeeper. The shopkeeper gave the balance money in coins of denomination Rs.2, Rs.5 and Rs.10. If these coins are in the ratio 5 ∶ 4 ∶ 1, then how many Rs.5 coins did the shopkeeper give?
A person divides a certain amount among his three sons in the ratio of 3 ∶ 4 ∶ 5. If he had divided this amount in the ratio of 1/3,1/4,1/5, his son, who had got the lowest share earlier, would get Rs.1,188 more. Find the amount (in Rs).
If a : b : c = \(\frac{1}{4} : \frac{1}{3} : \frac{1}{2}, \) then \( \ \frac{a}{b} : \frac{b}{c} : \frac{c}{a} = ?\)
A, B and C divide a certain sum of money among themselves. The average of the amount with them is Rs.4520. Share of A is \(10\frac{2}{3}%\) % more than share of B and \(33\frac{1}{3}%\) % less than share of C. What is the share of B (in Rs.)?
