Problem 422

Solution 1

Each of the four fractions is equal to the other three fractions, and each fraction has numerator 1, so then each denominator must be equal to the other three denominators.

That is, $a-2=b+2=c+1=d-3$.

If we let $b=4$, then $a-2=4+2=c+1=d-3$ or $a-2=6=c+1=d-3$, and so $a=8, c=5,$ and $d=9$.

Thus, the correct ordering is $b<c<a<d$.

Solution 2

Each of the four fractions is equal to the other three fractions, and each fraction has numerator 1, so then each denominator must be equal to the other three denominators.

That is, $a-2=b+2=c+1=d-3$, and by adding 3 to each, we get $a+1=b+5=c+4=d$.

Since $d=a+1$, then $d$ is one more than $a$ and so $a<d$.

Since $a+1=c+4$, then $a$ is 3 more than $c$ and so $c<a$.

Since $c+4=b+5$, then $c$ is 1 more than $b$ and so $b<c$.

Thus, the correct ordering is $b<c<a<d$.