Adding to 400
Find four integers whose sum is 400 and such that the first integer is equal to twice the second integer, three times the third integer and four times the fourth integer.
Age
14 to 16
Challenge level
Problem
Find four integers whose sum is $400$ such that the first integer is equal to twice the second integer, three times the third integer and four times the fourth integer.
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Answer: $192$, $96$, $64$ and $48$
first integer: $x$
second integer: $\frac12{x}$
third integer: $\frac13x\\$
fourth integer: $\frac14x$
$$\begin{align}x + \tfrac12x + \tfrac 13x + \tfrac 14x &= 400\\ \Rightarrow \tfrac{25}{12}x &= 400\\ \Rightarrow x &= 192\end{align}$$
Therefore the four numbers are, in order: $192$, $96$, $64$ and $48$.