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| Math Central | Quandaries & Queries |
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Question from Kenneth: Hello: 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? I am interested in this solution as to how I solved the problem. Mary is 3/2 of Sarah's age. I changed the denominators to 6th and added them to get 23/6. 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. 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.
Penny
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