Solving Age Word Problems with Ratios and Differences

This problem involves finding the age of a person given the age difference between two people and the ratio of their ages. We need to use algebraic equations to solve this.

Understanding the Given Information

We are given two key pieces of information about the ages of Amit and Arjun:

• Amit is younger than Arjun by 6 years. This tells us the difference in their ages.
• The ratio of Amit's age to Arjun's age is 5 : 7. This gives us a proportional relationship between their ages.

Setting up the Equations

Let's represent the ages of Amit and Arjun using variables:

• Let Amit's age be \(A\) years.
• Let Arjun's age be \(U\) years.

From the first piece of information, "Amit is younger than Arjun by 6 years", we can write an equation relating their ages:

$$U - A = 6$$

This can also be written as:

$$U = A + 6 \quad (*)$$

From the second piece of information, "the ratio of the ages of Amit and Arjun is 5 : 7", we can write another equation:

$$\frac{A}{U} = \frac{5}{7} \quad (**)$$

Solving for Amit's Age

Now we have a system of two linear equations with two variables. We can use substitution to solve for \(A\).

Substitute the expression for \(U\) from equation \((*)\) into equation \((**)\):

$$\frac{A}{A + 6} = \frac{5}{7}$$

To solve this equation, we can cross-multiply:

$$7 \times A = 5 \times (A + 6)$$

$$7A = 5A + 30$$

Now, we need to isolate the term with \(A\). Subtract \(5A\) from both sides of the equation:

$$7A - 5A = 30$$

$$2A = 30$$

Finally, divide both sides by 2 to find the value of \(A\):

$$A = \frac{30}{2}$$

$$A = 15$$

So, Amit's age is 15 years.

Verification

Let's check if this age fits all the conditions given in the problem.

• If Amit's age is 15, and Amit is 6 years younger than Arjun, then Arjun's age must be \(15 + 6 = 21\) years.
• The ratio of Amit's age to Arjun's age is \(15 : 21\).
• Simplifying the ratio \(15 : 21\) by dividing both numbers by their greatest common divisor, which is 3:
• \(\frac{15 \div 3}{21 \div 3} = \frac{5}{7}\)

The calculated ratio 5:7 matches the ratio given in the problem. The age difference \(21 - 15 = 6\) also matches the given difference. Thus, our calculated age for Amit is correct.

The age of Amit is 15 years.

Revision Table: Age Ratio Problem

Additional Information: Ratios and Proportions

Ratios are used to compare quantities. A ratio of 5:7 means that for every 5 units of the first quantity (Amit's age), there are 7 units of the second quantity (Arjun's age). In this problem, since their ages are in the ratio 5:7, we can say that Amit's age is \(5x\) and Arjun's age is \(7x\) for some common factor \(x\).

Using this approach, we can also solve the problem:

• Let Amit's age = \(5x\)
• Let Arjun's age = \(7x\)
• The difference in their ages is given as 6 years:
• Arjun's age - Amit's age = 6
• \(7x - 5x = 6\)
• \(2x = 6\)
• \(x = \frac{6}{2}\)
• \(x = 3\)

Now, substitute the value of \(x\) back into the expressions for their ages:

• Amit's age = \(5x = 5 \times 3 = 15\) years
• Arjun's age = \(7x = 7 \times 3 = 21\) years

This method also gives Amit's age as 15 years and confirms the difference \(21 - 15 = 6\).

Understanding ratios as parts of a whole or comparisons helps in setting up these types of problems. When dealing with age differences and ratios, setting up algebraic equations is a standard and effective approach.