Understanding Age Ratios and Differences

This problem involves calculating the difference in ages when given their ratio and sum. We are told that the ratio of Alpha's age to Beta's age is 2 ∶ 5. The sum of their ages is 238 years.

Let's represent their ages based on the given ratio. If the ratio is 2:5, we can say that Alpha's age is \(2x\) and Beta's age is \(5x\), where \(x\) is a common multiplier.

We know the sum of their ages is 238. So, we can write an equation:

Alpha's age + Beta's age = Sum of ages

\(2x + 5x = 238\)

Combining the terms on the left side, we get:

\(7x = 238\)

Now, we need to find the value of \(x\). We can do this by dividing the sum by the total parts in the ratio (2 + 5 = 7):

\(x = \frac{238}{7}\)

Let's perform the division:

\(238 \div 7\)

\(23 \div 7\) is 3 with a remainder of 2.

Bring down the 8, making it 28.

\(28 \div 7\) is 4.

So, \(x = 34\).

Now that we have the value of \(x\), we can find the actual ages of Alpha and Beta.

• Alpha's age = \(2x = 2 \times 34\)
• Alpha's age = \(68\) years
• Beta's age = \(5x = 5 \times 34\)
• Beta's age = \(170\) years

To verify, let's check the sum of their ages: \(68 + 170 = 238\), which matches the given information.

The question asks for the difference between their ages. The difference is calculated by subtracting the younger age from the older age.

Difference = Beta's age - Alpha's age

Difference = \(170 - 68\)

Difference = \(102\)

So, the difference between Alpha's and Beta's ages is 102 years.

Revision Table: Key Steps for Ratio Age Problems

Additional Information: Understanding Ratios in Word Problems

A ratio is a comparison of two or more quantities. In this case, the ratio 2:5 means that for every 2 units of age Alpha has, Beta has 5 units. When solving ratio problems involving sums or differences, it's often helpful to think of the quantities as multiples of a common value, represented by a variable like \(x\).

The sum of the ratio parts (2 + 5 = 7) tells us that the total sum of the ages is divided into 7 equal parts, each part being equal to the common multiplier \(x\). Finding \(x\) allows us to scale the ratio parts back to the actual values of the ages.

Ratio problems can involve various operations:

• Sum: Given the ratio and the sum of quantities, find the quantities. (Like this problem)
• Difference: Given the ratio and the difference between quantities, find the quantities.
• Individual Value: Given the ratio and the value of one quantity, find the other quantities or the sum/difference.

Understanding how to use a common multiplier \(x\) is key to solving these types of problems algebraically.