Question

$x+y+z=850$. If $x$ is reduced by $100$, $y$ by $25$, and $z$ by $50$, then $(x-100):(y-25)=1:2$ and $(y-25):(z-50)=5:6$. Find the original value of $x+y$.

• A. 400
• B. 500
• C. 550
• D. 350

Hint:

When ratios involve shifted values (like $x-100$), set them equal to $k$-multiples, back-substitute in the sum, and solve.

Answer 1

The Correct Option is B

Step 1: Convert ratios to algebra.
Let $x-100=1k,\; y-25=2k \(\Rightarrow\) x=k+100,\; y=2k+25.$
From $(y-25):(z-50)=5:6$, let $y-25=5m,\; z-50=6m$. But $y-25=2k$, so $2k=5m \(\Rightarrow\) m=\frac{2k}{5}$. Hence $z=50+6m=50+\frac{12k}{5}$.
Step 2: Use $x+y+z=850$.
$(k+100)+(2k+25)+\left(50+\frac{12k}{5}\right)=850$
$\Rightarrow \frac{27k}{5}+175=850 \(\Rightarrow\) \frac{27k}{5}=675 \(\Rightarrow\) 27k=3375 \(\Rightarrow\) k=125.$
Step 3: Find $x$ and $y$ and sum.
$x=125+100=225,\; y=2.....125+25=275 \(\Rightarrow\) x+y=225+275=500.$
\boxed{\text{Original }x+y=500}