Math Word Problems: Solving Age Problems Part 3

This is the third part of the Solving Age Problems of the Math Word Problem Solving Series. In this post, we discuss more complicated age word problems.

Problem 7

Anna who is 6 years old and his father Ben who is 27 years old have the same birthday. In how many years will Ben be twice as old as Anna?

Solution

As years go by, the number of years added to Ben’s and Anna’s ages is the same. If we let the number of years that have gone by be x, then in x years, their ages will be

Ben’s Age: 27 + x

Anna’s Age: 6 + x

Since in x years, Ben will be as twice as old as Anna, if we multiply Anna’s age by 2, their ages will be equal. So, we can now set up the equation

2(6 + x) = 27 + x.

Solving, we have 12 + 2x = 27 + x. which means that x = 15.

That means that in 15 years, Ben will be twice as old as Anna.

Checking the Answer

In 15 years, Ben will be 27 + 15 = 42 years old. In 15 years, Anna will be 6 + 15 = 21 years old. Clearly, Ben will be twice as old as Anna in 15 years, so we are correct.

Problem 8

The sum of the ages of James and Clark is 50. Five years ago, Clark’s age was three times as James’ age. How old are James and Clark now?

Solution

If James is x years old, then Clark is 50 - x years old (Why?). Five years ago, their ages were

James’ age: x - 5

Clark’s age: (50 - x) - 5 = 45 - x.

Since 5 years ago, Clark’s age was three times as that of James, if we multiply James’ age by 3, their ages will be equal. So, we set up the equation

3(x-5) = 45-x

Solving we have, 3x - 15 = 45 - x. This means that 4x = 60 and x = 15. Therefore, James is 15 years old and Clark is 35 years old.

Checking the Answer

The sum of James age which is 15 and Clark’s age which is 35 is equal to 50. Five years ago, James was 10 years old and Clark was 30. As we can see, Clark was three times as old as James. Therefore, we are correct.

Problem 9*

Pol is 10 years younger than Greg. In 7 years, he will be 10 years more than one half as old as Greg. Find their age at present.

Solution

Let x be Greg’s age and  let x - 10 be Pol’s age.  In 7 years, their ages will be

Greg’s age: x + 7

Pol’s age: x - 10 + 7 = x - 3.

In the problem, it is said that in 7 years, Paul’s age will be ten years more than half of Greg’s age. Now, in 7 years, half of Greg’s age will be

\frac{1}{2}(x + 7).

Now, since Paul is 10 more than one half Greg’s age, if we add 10 years to half Greg’s age which is \frac{1}{2}(x + 7), their ages will be equal. Therefore,

\frac{1}{2}(x + 7) + 10 = x - 3.

Multiplying both sides by 2 we have x + 7 + 20 = 2x - 6. Simplifying, we have x + 27 = 2x - 6 which gives us x = 33. Therefore, Greg is 33 years old and Pol is 23.

Checking the Answer

Greg is 33 and Paul is 23, so Pol is 10 years younger. In 7 years, their ages will be 40 and 30 respectively. Half of Greg’s age by then is 20 and if we add 10, the result is 30 which equal’s Pol’s age.

*Source: http://algebra-word-problems.blogspot.com/2010/10/age-problem-5.html

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