Question

The sum of X, Y, W, and Z is 84. The sum of Y, W, and Z is three times X. The sum of X, W, and Z is 320% of Y. W is 20% of the sum of X, Y, and Z. Find the value of Z.

• A. 29
• B. 42
• C. 32
• D. 40

Hint:

Convert percentage statements into algebraic equations and solve systematically.

Answer 1

The Correct Option is A

Step 1: Use the first relation.
We are given $Y+W+Z=3X$ and the total $X+Y+W+Z=84$. Substituting the first into the second gives $X+3X=84$, so $4X=84$ and \[ X=21 \]
Step 2: Set up the second relation.
Also $X+W+Z=3.2Y$, and from the total $X+W+Z=84-Y$.
Step 3: Solve for Y.
Equating, $84-Y=3.2Y$, so $84=4.2Y$ and \[ Y=20 \]
Step 4: Express W from the third relation.
Since $W$ is $20\%$ of $X+Y+Z$, \[ W=\frac{1}{5}(21+20+Z)=\frac{41+Z}{5} \]
Step 5: Use the total to link W and Z.
With $X=21$ and $Y=20$, the total gives $W+Z=84-41=43$.
Step 6: Solve for Z.
Substitute $W$: \[ \frac{41+Z}{5}+Z=43 \implies 41+6Z=215 \implies 6Z=174 \implies Z=29 \]
\[ \boxed{29} \]