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You are here: Experts > Science > Math for Kids > Word Problems > Fractions
Expert: Scott A Wilson - 11/1/2009
Question
QUESTION: Hello:
I sent you this question recently, and I have a follow-up question that follows your answer to my first question.
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2/3 of Mary’s age equals Sarah’s age and 3/4 of Ruth’s age equals Sarah’s age. If the total ages equal 46, how old is each girl?
Answer: Mary 18, Ruth 16, Sarah 12
Solution:
3/2 of Sarah’s age equals Mary’s age.
4/3 of Sarah’s age equals Ruth’s age.
6/6 of Sarah’s age equals Sarah’s age.
46 divided by 23/6 equals Sarah’s age of 12.
Can the following similar situation be solved if the two girls ages do not equal Sarah’s age?
2/3 of Mary’s age equals Ruth’s age and 3/4 of Ruth’s age equals Sarah’s age. How old is each girl?
Their ages may not be the same as those in the above example.
I thank you for your reply.
----------------------- Answer
2M/3 = R, 3R/4 = S, so putting R in gives 3(2M/3)/4 = S, so M/2 = S, so M = 2S.
Also, it can be seen that R = 4S/3.
The sum of the ages is still 46.
Since this time we have M+R+S=46, we can put in what M and R are.
This says 2S + 4S/3 + S = 46, so 6S/3 + 4S/3 + 3S/3 = 46.
this gives 13S/3 = 46, or S = 3*46/13, or S = 138/13 = 10 8/13.
R is (4/3)S = (4/3)(138/13) = 184/13 = 14 2/13.
M is 2S = 2(138/13) = 276/13 = 21 3/13.
M+R+S=(276+184+138)/13 = 598/13 = 46.
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Can the same method for determining the ages in the above examples be used to determine the ages of the girls in the example below?
1/2 of Mary's age and 2/5 of Ruth's age and 1/4 of Mary's age equal 14.
How old is each girl?
Answers: 10, 15, 12
I thank you for your reply.
ANSWER: If Mary is M and Ruth is R, is this to say that M/2 + 2R/5 + M/4 = 14?
That would say that 3M/4 + 2R/5 = 14.
Solve the equation for M. This gives 3M/4 = 14 - 2R/5, or M = 4(14 - 2R/5)/3.
This can be changed to M = 56 - 2R/15.
This means is R was 15, M would be 54.
If R was 30, M would be 52.
If R was 45, M would be 50.
If R was 60, M would be 48.
Perhaps the last M is suppose to be an S.
This makes the equation M/2 + 2R/5 + S/4 = 14.
Again, there are multiple solutions.
The equation to use is M = 2(14 - 2R/5 - S/4).
If R was 5, and S was 4, then M would be 22.
If R was 10, and S was 12, then M would be 14.
If R was 15 and S was 8, then M would be 12.
Do you mean to include something else?
---------- FOLLOW-UP ----------
QUESTION: Hello:
I want to thank you for your reply.
Here is the complete calculation: 1/2 of Mary's age [10] and 2/5 of Ruth's age [15] and 1/4 of Mary's age [12] equals 14.
1/2 X 10 = 5
2/5 X 15 = 6
1/4 X 12 = 3
5 + 6 + 3 = 14
The missing ages are in the brackets. Can you determine these ages by using the solution similar to the first example?
I thank you for your follow-up reply.
Answer
I assume second Mary is supposed to be a different person.
After all, the first one is 10 and the second one is 12.
Using Excel, I found the following possibilites for the ages:
M R X
8 15 16 8/2 + 2*15/5 + 16/4 = 4 + 6 + 4 = 14
10 15 12 10/2 + 2*15/5 + 12/4 = 5 + 6 + 3 = 14
12 10 16 12/2 + 2*10/5 + 16/4 = 6 + 4 + 4 = 14.
12 15 8
14 10 12
14 15 4
16 5 16
16 10 8
18 5 12
18 10 4
The caluclations were shown for the 1st 3.
Given the equation M/2 + 2R/5 + X/4, the others work as well.
Note that 10, 15, 12 is the second choice, but there are 9 others that also work.
There are even more solutions, but I kept M each to an integer in the equation.
One of the other ones is M 17, R 5, X 14, for 17/2 + 2*5/5 + 14/4 = 8.5 + 2 + 3.5 = 14.
There are probably somewhere around 20 of them like this.
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