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 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 years old, then Bill is years old. Now, one fifth of Bill’s age is and one half of Carol’s age is . Add these together and you get . Now, we have an equation.

**Solution**

Let be Carol’s age and be Bill’s age.

.

Simplifying, we have

.

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

.

.

This means that Carol is and Bill is .

**Check**

Bill is and Carol is . Yes, Bill is four times as old as Carol. One fifth of is . One half of is and . 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 years old, then the kid is Six years from now, the ages of the mother and the kid respectively are and as shown in the table below.

As the kid said, in 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

.

Now, we write the solution.

**Solution**

Let be the mother’s age and be the kid’s age.

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

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

Again, by distributive property, we have

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

.

So, the mother and and the kid is . A smart kid indeed, giving problems such as this at age 6.

*Check*

Left as an exercise.

**Example 9**

Donna is 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 years old and Donna is . Now, Donna’s age a year a go is which is equal to . How, one fifth of Donna’s age a year ago is and one fourth of Demi’s age is .

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

Solution

Let be Demi’s age and be Donna’s age

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

.

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

Therefore, Demi is and Donna is .

Check: Left as an exercise.

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?

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

Naguguluhan din po ako. One third po diba is 1/3 is not the same as to 3.. Bakit po sa 3 minultiply? Thanks po.

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

is also

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

Ask ko lang po dun kay Carol and Bill, ung 13×10 is 130. Then sa baba naging 30 na lang?

nagkamali lang po siguro. kasi kung titignan po yung solution, 30/13 is no longer equal to 10 po. mis-type lang siguro po