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Question from Kenneth:


Here is my word problem question:

Sarah's age is 2/3 of Mary's age and 3/4 of Ruth's age. The sum of their ages is 46 years. How old is each?
Answers: Sarah-12, Mary-18, Ruth-16

I am interested in this solution as to how I solved the problem.

Mary is 3/2 of Sarah's age.
Ruth is 4/3 of Sarah's age.
Sarah is 6/6 of her own age.

I changed the denominators to 6th and added them to get 23/6.
I divided 46 by 23/6 and got Sarah's age of 12.

Is it possible to solve this similar word problem by using the same method?

Sarah's age is 2/3 of Mary's age and 1 & 1/8 of Ruth's age is Mary's age.
The total of their ages is 46. How old is each?

Please read the question carefully. It is very similar to the first except Ruth's age is compared to Mary's age not Sarah's.

I thank you for your assistance and reply.

Hi Kenneth,

Your solution is fine and with a little care the technique can be used to solve similar problems. You have

Mary is 3/2 of Sarah's age.
Ruth is 4/3 of Sarah's age.
Sarah is 6/6 of Sarah's age.

(I added some emphasis and changed the wording of the last statement.) What is important is that you have selected one girl, Sarah, and expressed each girl's age as a fraction of Sarah's age. A more conventional algebraic way to write this is to let M, R and S be the three girl's ages and then your statements are

M = 3/2 S
R = 4/3 S
S = 6/6 S

You then expressed the three fractions with the same denominator and added the three equations to get

M + R + S = 23/6 S

which you know is 46 so 23/6 S = 46 and hence S = 12.

For your second problem you have

Sarah is 2/3 of Mary's age
9/8 of Ruth is Mary's age

Mary's age is common in the two statements, which is why I wrote it on the right, so to make this work I need to express Ruth's age as a fraction of Mary's age. If 9/8 of Ruth's age is Mary's age then 8/9 of Mary's age is Ruth's age. Thus the three statements (equations) become

Sarah is 2/3 of Mary's age
Ruth is 8/9 of Mary's age
Mary is 9/9 of Mary's age

Expressing each fraction with a denominator of 9 gives a sum of 23/9. If the sum is of the ages is 46 then 23/9 of Mary's age is 46 so Mary's age is 18.


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