The ratio of boys and girls in a group is 7 : 6. If 4 more boys join the group and 3 girls leave the group, then the ratio of boys to girls becomes 4 : 3. What is the total number of boys and girls initially in the group?
104
Solving the Ratio of Boys and Girls Problem
This problem involves ratios and setting up an algebraic equation to find the initial number of boys and girls in a group. We are given the initial ratio, the changes in the number of boys and girls, and the resulting new ratio.
Setting Up the Initial Numbers
The initial ratio of boys to girls is given as 7 : 6.
We can represent the initial number of boys and girls using a common multiple. Let this multiple be \(x\).
- Initial number of boys = \(7x\)
- Initial number of girls = \(6x\)
The total number of boys and girls initially in the group is the sum of the initial number of boys and girls, which is \(7x + 6x = 13x\).
Applying the Changes to the Group
The problem states that 4 more boys join the group and 3 girls leave the group.
- New number of boys = Initial boys + 4 = \(7x + 4\)
- New number of girls = Initial girls - 3 = \(6x - 3\)
Forming the New Ratio and Equation
After the changes, the ratio of the new number of boys to the new number of girls becomes 4 : 3.
We can write this as an equation:
\(\frac{\text{New number of boys}}{\text{New number of girls}} = \frac{4}{3}\)
\(\frac{7x + 4}{6x - 3} = \frac{4}{3}\)
Solving the Equation for x
To solve for \(x\), we can cross-multiply:
\(3(7x + 4) = 4(6x - 3)\)
Now, distribute the numbers on both sides of the equation:
\(21x + 12 = 24x - 12\)
Gather the \(x\) terms on one side and the constant terms on the other side. Subtract \(21x\) from both sides:
\(12 = 24x - 21x - 12\)
\(12 = 3x - 12\)
Add 12 to both sides:
\(12 + 12 = 3x\)
\(24 = 3x\)
Divide by 3 to find the value of \(x\):
\(x = \frac{24}{3}\)
\(x = 8\)
Calculating the Initial Total Number
We found that \(x = 8\).
The initial number of boys was \(7x\), so initial boys = \(7 \times 8 = 56\).
The initial number of girls was \(6x\), so initial girls = \(6 \times 8 = 48\).
The total number of boys and girls initially in the group is \(13x\).
Total initial number = \(13 \times 8\)
Total initial number = \(104\)
Alternatively, we can add the calculated initial number of boys and girls:
Total initial number = Initial boys + Initial girls = \(56 + 48 = 104\).
The total number of boys and girls initially in the group is 104.
Let's check if these initial numbers satisfy the conditions:
- Initial ratio: \(56 : 48\). Divide both by 8: \(7 : 6\). (Correct)
- After changes: Boys = \(56 + 4 = 60\), Girls = \(48 - 3 = 45\).
- New ratio: \(60 : 45\). Divide both by 15: \(4 : 3\). (Correct)
The calculated initial numbers satisfy both ratio conditions.
Revision Table: Understanding Ratio Problems
| Concept | Description | Example Application |
|---|---|---|
| Ratio | A comparison of two or more quantities of the same kind, usually expressed as a fraction or using a colon (e.g., a:b). | Boys to girls ratio is 7:6. |
| Representing Ratio with Variable | If a ratio is a:b, quantities can be represented as ax and bx, where x is a common multiplier. | Initial boys = 7x, Initial girls = 6x. |
| Forming Equation from New Ratio | After changes, set up a new ratio expression and equate it to the given new ratio. | \(\frac{7x+4}{6x-3} = \frac{4}{3}\) |
| Solving Linear Equation | Use algebraic techniques (like cross-multiplication, combining like terms) to isolate the variable. | Solve \(3(7x+4) = 4(6x-3)\) for x. |
| Finding Initial Quantities | Substitute the found variable value back into the initial expressions (ax and bx). | Calculate 7x and 6x using the value of x. |
Additional Information: Key Concepts in Ratio and Algebra
This problem effectively combines the concept of ratios with basic algebraic equation solving. Understanding these concepts is crucial for quantitative aptitude problems.
What is a Ratio?
A ratio shows how many times one number contains another or how many times a quantity is contained in another. It's a way to compare the relative sizes of two or more values. Ratios can be simplified by dividing all terms by the same common factor, just like fractions.
Setting up Equations from Word Problems
Translating word problems into mathematical equations is a fundamental skill. For ratio problems involving changes, it's often helpful to represent the initial quantities using a variable (\(x\)) based on the given ratio. Then, express the new quantities after the changes in terms of \(x\). Finally, use the new ratio to set up an equation that can be solved for \(x\).
Solving Linear Equations
The equation we solved, \(3(7x + 4) = 4(6x - 3)\), is a linear equation. Solving it involves a few standard steps:
- Remove parentheses by distributing.
- Combine like terms on each side of the equation.
- Move terms with the variable to one side and constant terms to the other using addition or subtraction.
- Isolate the variable by dividing or multiplying.
Practicing these steps will help you solve various algebraic problems derived from word problems.
Important Questions from Direct or Indirect Proportion
If 3A = 4B = 5C, then A : B : C is equal to:
The monthly incomes of A and B are in the ration 3 : 5 and the ratio of their savings is 2 : 3 If the income of B is equal to three times the savings of A, then what is the ratio of the expenditures of A and B?
If an amount of Rs. 990 is divided among A, B and C in the ratio of 3 : 4 : 2, then B will get:
The ratio of the number of boys to the number of girls in a school of 640 students, is 5 : 3. If 30 more girls are admitted in the school, then how many more boys should be admitted so that the ratio of boys to that of the girls, becomes 14 : 9.
In a wallet, there are notes of the denominations Rs. 10 and Rs. 50. The total number of notes is 12. The number of Rs. 10 and Rs. 50 notes are in the ratio of 1 : 2. Total money in the wallet is:
