How to Solve Age Problems Part 3

This is the third part of the Solving Age Problems Series. In this part, we will solve age problems with a variety of formats and difficulty that are not discussed in the first two parts. We have already solved six problems in the first and second part, so we start with the seventh problem.

Example 7

Bill is four times as old as Carol. One fifth of Bill’s age added to one half Carol’s age is equal to 13 years. How old are both of them?

Scratch work

Bill is older than Carol and he is four times older. This means that if Carol is x years old, then Bill is 4x years old. Now, one fifth of Bill’s age is \frac{1}{5}(4x) and one half of Carol’s age is \frac{1}{2}x. Add these together and you get 13. Now, we have an equation.

Solution

Let x be Carol’s age and 4x be Bill’s age.

\frac{1}{5}(4x) + \frac{1}{2}x = 13.

Simplifying, we have

\frac{4}{5}x + \frac{1}{2}x = 13.

Since we have a fraction, we can eliminate the denominator by multiplying everything with the least common multiple of 5 and 2 which is 10. Multiplying both sides of the equation by 10, we have

\displaystyle \frac{40}{5}x + \frac{10}{2}x = 130.

\displaystyle 8x + 5x = 30

13x = 30

x = 10.

This means that Carol is 10 and Bill is 40.

Check

Bill is 40 and Carol is 10. Yes, Bill is four times as old as Carol. One fifth of 40 is 8. One half of 10 is 5 and 8 + 5 = 13. So, we are correct.

Example 8

When a really smart math kid was asked about his age, he said:

“I am one fifth as old as my mother. In six years, I will be one-third as old.”

How old is the kid and his mother?

Scratch Work

The kids is one fifth as old as his mother. So, if the mother is x years old, then the kid is \frac{1}{5}x Six years from now, the ages of the mother and the kid respectively are x + 6 and \frac{1}{5}x + 6 as shown in the table below.

Screen Shot 2014-03-21 at 3.52.25 PM

 

As the kid said, in 6 years, his age will be a third of his mother. This means that if we multiply his age by $latex3$, then it will equal the age of his mother. In equation form, we have

3(\frac{1}{5}x + 6) = x + 6.

Now, we write the solution.

Solution

Let x be the mother’s age and \frac{1}{5}x be the kid’s age.

x + 6 = 3(\frac{1}{5}x + 6)

We simplify the right hand side by Distributive Property. This gives us

x + 6 = \frac{3}{5}x + 18

Now, to eliminate the fraction, we multiply both sides of the equation by 5.

5(x + 6) = 3x + 90

Again, by distributive property, we have

5x + 30 = 3x + 90

Putting all the x’s on the left hand side and all the numbers on the right hand side, we have

5x - 3x = 90 - 30

2x = 60

x = 30.

So, the mother and 30 and the kid is \frac{1}{5}(30) = 6. A smart kid indeed, giving problems such as this at age 6.

Check

Left as an exercise.

Example 9

Donna is 6 years older than Demi. One fifth of Donna’s age a year ago added to three fourth of Demi’s age is equal to Demi’s age. How old is Donna?

Scratch Work

Demi is x years old and Donna is x + 6. Now, Donna’s age a year a go is x + 6 - 1 which is equal to x + 5. How, one fifth of Donna’s age a year ago is \frac{1}{5}(x+5) and one fourth of Demi’s age is \frac{1}{4}x.

Now, these ages if added equal’s Donna’s age which is x. Therefore, the equation is

\frac{1}{5}(x + 5) + \frac{3}{4}x = x

Solution

Let x be Demi’s age and x + 6 be Donna’s age

\frac{1}{5}(x +5) + \frac{3}{4}x = x

Simplifying, the left hand side by distributive property, we have

\frac{1}{5}x + 1 + \frac{3}{4}x = x.

Now, to eliminate the fractions, we multiply both sides of the equation by the least common multiple of 5 and 4 which is 20. This will result to

\frac{20}{5}x + 20 + \frac{60}{4}x = 20x

4x + 20 + 15x = 20x

19x + 20 = 20x

20 = x

Therefore, Demi is 20 and Donna is 26.

Check: Left as an exercise.

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3 Responses

  1. Michael says:

    Hi! Ask ko lang po ‘yung sa Example 8:

    “I am one fifth as old as my mother. In six years, I will be one-third as old.”

    Typo-error lang po ba ‘yung “one-third” as old in six years?

    • Civil Service Reviewer says:

      Hindi po yan typo, talagang ganyan yan, kaya minultiply ng 3 yung age ng bata para maging equal sila.

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