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: Algebraic Representation of Ratios
Given \(x-100=1k,\; y-25=2k\), which implies \(x=k+100,\; y=2k+25.\)
From \((y-25):(z-50)=5:6\), let \(y-25=5m,\; z-50=6m\). Since \(y-25=2k\), then \(2k=5m \Rightarrow m=\frac{2k}{5}\). Therefore, \(z=50+6m=50+\frac{12k}{5}\).
Step 2: Utilize the Equation \(x+y+z=850\)
Substitute the expressions for \(x, y,\) and \(z\) into \(x+y+z=850\): \((k+100)+(2k+25)+\left(50+\frac{12k}{5}\right)=850\)
Simplifying: \(\frac{27k}{5}+175=850 \Rightarrow \frac{27k}{5}=675 \Rightarrow 27k=3375 \Rightarrow k=125.\)
Step 3: Calculate \(x, y\), and their Sum
Calculate \(x\) and \(y\): \(x=125+100=225,\; y=2 \cdot 125+25=275\), and find their sum: \(x+y=225+275=500.\)
\boxed{\text{Original }x+y=500}