# How Many Hearts? John and Hanna designed together 60 valentines. If we divide the number representing valentines designed by John by the number representing valentines designed by Anna, the quotient is 3.

How many combinations of the two numbers are possible?

Answer

Let j be the number of valentines designed by John, and let h be the number of valentines designed by Hanna. We have to find all the possible combinations (j, h) such that j + h = 60, and the quotient of j divided by h is 3.

Can we say that 3h = j? Yes, but it would not be a complete assessment.

We know that the quotient from dividing j by h is 3, but what is the remainder? The problem does not state that j divides by h evenly. The remainder could be 0, 1, or 2.

Therefore, j could also be greater than 3h, or 3h < j . How much greater? Can j = 4h? No, because in this case the quotient would be 4, not 3.

So we can write that 3h ≤ j < 4h.

Adding h to each side of the inequality, we get 3h + h ≤ j + h < 4h + h.

4h ≤ j + h < 5h

Since j + h = 60, we have

4h ≤ 60 < 5h, and therefore 12 < h ≤ 15.

All the possible combinations (j, h) are (45, 15), (46, 14), and (47, 13).

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