The ratio of the number of marbles received by John and Peter was 4:7 respectively.
The ratio of the number of marbles received by Peter and Sam was 9:5. John gave
1/12 of his marbles to Sam. Peter gave 1/9 of his marbles to Sam. Eventually, Sam
had 135 marbles.
(a) Find the ratio of the number of John's marbles to Sam's marbles at first.
(b) Find the total number of marbles received by the 3 boys.
Ratios can be scaled up or down by multiplication or division.
So for example the ratio 4 : 7 is the same as 8 : 14 (having multiplied by 2), or 12 : 21 ( having multiplied by 3) and so on.
With problems like this, where two ratios are involved, the thing to do is to arrange for two of the components, (in different ratios) to be equal to each other. So, multiply the first ratio by 9 and the second by 7 to arrive at
J : P = 36 : 63, and P : S = 63 : 35.
The two can then be shunted together so J : P : S = 36 : 63 : 35.
The actual number of marbles for each person will then be (for some positive integer k), 36k for John, 63k for Peter and 35k for Sam.
John then gives 3k and Peter gives 7k to Sam.
That means that Sam now has 35k + 3k + 7k = 45k marbles and that has to equal 135, so k = 3, etc. .
Ratio of marbles of Peter by Sam, P : S = 9 : 5
Now, J : P : S = 36 : 63 : 35
(a) Ratio of John by Sam : J : S = 36 : 35
(b) Let the number of marbles with John = 36x
Peter = 63x
Sam = 35x
When John Gabe 1/12 of his marble to Sam and
Peter gave 1/9 of his marbles to Sam. Then
marbles with Sam = 35x + (36x * 1/12)
+ (63 * 1/9)
= 35x + 3x + 7x
According to Question
45x = 135
=> x = 135/45 = 3
Total marbles + (36x + 63x + 35x) = 134x = 134 * 3