Expert: Ahmed Salami - 2/5/2006

Question
Hello:

I want thank you for the reply. Can the following calculation work?

Sarah's age is 6/6 of itself. So, 6/6 is 2/3 of Mary's age and 6/6 is 3/4 of Ruth's age.

6/6 is 2/3 of 18/12. 6/6 is 3/4 of 24/18. I do not know what to do with these results. Do you have any ideas?

I thank you for your help and assistance.



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Followup To
Question -
Hello:

Sarah's age is 2/3 of Mary's and 3/4 of Ruth's. The sum of their ages is 46 years. How old is each?

NOTE: Find an expression for each in sixths of Sarah's age.

Answers: 46 divided by 23/6 = 12, 46 divided by 23/9 = 18, and 46 divided by 23/8 = 16.
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The question came from a chapter of an old textbook on arithmetic. The chapter pertains to fractions and makes no mention of algebra calculations. The solution should not be determined by using algebra.

I believe the solution is something as follows:
46 divided by 6/6 + 3/2 + 4/3 = 46 divided by 6/6 + 9/6 + 8/6 equals 46 divided by 23/6 equals 12, Sarah's age.

The 6/6 is used to indicate Sarah's age in the total amount for their ages of 46, 6/6 of Sarah's + 2/3 of Mary's + 3/4 of Ruth's. It's their ages that total 46 not the sum of these calculations. Their denominators are then changed to sixths so they can be added together. The sum of 6/6 + 9/6 + 8/6 equals 23/6. The tricky part is Sarah's age being 6/6 of her own age and knowing to add this into the calculation.

Here is my question:

I am curious to know why invert the fractions 2/3 and 3/4 before adding them and then dividing. When 46 is divided by the fractions, they are inverted again, and then 46 is multiplied by them. Do you have any ideas or an explanation for inverting these fractions first and then dividing which again requires inverting them? In other words, what is the logic used in the solution?

I thank you for your comments or explanation.

Answer -
Hi,
Sorry for the time it took. Technical problem with the site.
The simple idea of inverting comes from the knowledge
that if A = xB, then B = A/x
1/x being the multiplicative inverse of x
Now, if Sarah's age S is (2/3)Mary's age M, then
S = (2/3)M
and M = (3/2)S
If Sarah's age is (3/4)Ruth's age,
S = (3/4)R
and R = (4/3)S
The sum of their ages,
S + M + R = S + (3/2)S + (4/3)S
46 = (23/6)S
S = 12
M = (3/2)12 = 18
R = (4/3)12 = 16
I hope i have helped. You can always get back to me.
Regards.

Answer
Hi,
I think you're doing great. All you need add is that in terms of Sarah's age, the sum of the ages would be
6/6 + 18/12 + 24/18
= 1 + 3/2 + 4/3 = 23/6
The sum of the ages would then be Sarah's age multiplied by 23/6, which of course leads you to the solution.

Regards.