2500.41 – Milly and Molly Make Muffins. So Do Their Mothers!

Milly made one more muffin that Molly’s mother and Milly’s mother made one more muffin than Molly made. Milly and Molly and their mothers together made 50 muffins. At the same time, Milly and Molly’s mother together made 4 muffins more that Molly’s and Milly’s mother together made. How many muffins were made by Milly?

Solution

Too many M’s! Let’s assign variables as follows:

  • A=A = Milly’s muffins

  • B=B = Molly’s muffins

  • C=C = Milly’s mother’s muffins

  • D=D = Molly’s mother’s muffins.

We want AA. We are given four equations:

  • A=D+1A = D + 1

  • C=B+1C = B + 1

  • A+B+C+D=50A + B + C + D = 50

  • A+D=B+C+4A + D = B + C + 4

Using the first two equations we can eliminate AA and CC in the second two equations:

A+B+C+D=50(D+1)+B+(B+1)+D=502B+2D+2=50B+D=24\begin{align*}A + B + C + D &= 50 \\(D + 1) + B + (B + 1) + D &= 50 \\2B + 2D + 2 &= 50 \\B + D &= 24\end{align*}

and

A+D=B+C+4(D+1)+D=B+(B+1)+42D+1=2B+5D−B=2\begin{align*}A + D &= B + C + 4 \\(D + 1) + D &= B + (B + 1) + 4 \\2D + 1 &= 2B + 5 \\D - B &= 2\end{align*}

We are left with two equations in two variables. Piece of cake!

B+D=24−B+D=2\begin{align*}B + D &= 24 \\-B + D &= 2\end{align*}

Therefore, 2D=262D = 26 and D=13D = 13. Tracing on backwards, B=10B = 10, so C=11C = 11 and A=14A = 14. So Milly made 14 muffins.