How many marbles did Mary buy?

This is a challenging question that we came across in class today. Here’s the question and two possible methods. Let me know if you have other methods!

Question:
Mary bought some red marbles and gave half to Noel.
Noel bought some blue marbles and gave half to Mary.
Mary lost 16 red marbles and Noel lost 55 blue marbles.
The ratio of Mary’s red marbles to blue marbles became 18:85.
The ratio of Noel’s red marbles to blue marbles became 7:20.
How many red marbles did Mary buy?

The solution is to relate the before and after together.
Recognise that Mary and Noel have the same number of red marbles. They also have the same number of blue marbles. But the number of red marbles may not be the same as the blue marbles.

Method 1.
Given the above, we can use “units” or “u” for red marbles, and “parts” or “p” for blue marbles.

So in the beginning M and N has 1u red each, and 1p blue each. After M lost 16 red, and N lost 55 blue, they have:
M: (1u-16) red and (1p) blue
N: (1u) red and (1p-55) blue
this is 18 : 85
and 7: 20

cross multiply we have:
85 x ( 1u-16) = 18p ————(A)
and
20 x (1u) = 7 x (1p-55) ———— (B)

from A,
85u = 18p+ 1360
340u = 72p + 5440

From B,
20u = 7p – 385
340u = 119p – 6545

together:
72p + 5440 = 119p – 6545
so 47p = 11985
p = 255
u = 70
since Mary bought 2u of red marbles, 2u = 140

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Method 2.
We can work from the back. After they lost the marbles, the ratio is:
M-> 18 : 85
N-> 7 : 20

Before they lost the marbles:
M-> 18u + 16 : 85u
N-> 7p : 20p + 55

We know that before they lost the marbles, they have the same red, and the same blue. So:
18u + 16 = 7p ———-(A)
85u = 20p + 55 ————(B)

Again we can try to solve this.
From (B)
17u = 4p + 11
119u = 28p + 77

From (A)
28p = 72u + 64

Put them together:
119u = (72u + 64) + 77
47u = 141
u = 3

We know mary bought twice of 18u + 16
= 2 x (18 x 3 + 16) = 2×70 = 140