The sum of three numbers is 98. The first number is 2/3 of the second, and the second is 5/8 of the third. What is the second number?
Hello, Amy!
The sum of three numbers is 98.
The first number is 2/3 of the second, and the second is 5/8 of the third.
What is the second number?
Let the three numbers be: $\displaystyle a,b,c.$
"Their sum is 98": .$\displaystyle a + b + c \:=\:98$ .[1]
"The first is 2/3 of the second": .$\displaystyle a \:=\:\frac{2}{3}b$ .[2]
"The second is 5/8 of the third": .$\displaystyle b \:=\:\frac{5}{8}c \quad\Rightarrow\quad c \:=\:\frac{8}{5}b$ .[3]
Substitute [2] and [3] into [1]: .$\displaystyle \frac{2}{3}b + b + \frac{8}{5}b \:=\:98$
Multiply by 15: .$\displaystyle 10b + 15b + 24b \:=\:1470 \quad\Rightarrow\quad 49b \:=\:1470$
. . Therefore: .$\displaystyle \boxed{b \;=\;30}$