# 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

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

$\displaystyle 8x + 5x = 130$

$13x = 130$

$x = 10$.

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.

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.

### 1 Response

1. OMS says:

Ang sabi is Donna’s age “a year ago” so dapat meron minus 1 sa formula.