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 Conversion of Ratios
Define \(x-100=1k\) and \(y-25=2k\), which implies \(x=k+100\) and \(y=2k+25\). From \((y-25):(z-50)=5:6\), let \(y-25=5m\) and \(z-50=6m\). Since \(y-25=2k\), then \(2k=5m\), thus \(m=\frac{2k}{5}\). Therefore, \(z=50+6m=50+\frac{12k}{5}\).

Step 2: Utilize the Equation \(x+y+z=850\)
Substitute the derived expressions into \(x+y+z=850\): \((k+100)+(2k+25)+\left(50+\frac{12k}{5}\right)=850\). Simplifying, \(\frac{27k}{5}+175=850\), which leads to \(\frac{27k}{5}=675\) and subsequently \(27k=3375\). Solving for \(k\), we get \(k=125\).

Step 3: Calculate \(x\), \(y\), and their Sum
Calculate \(x\) and \(y\) using \(k=125\): \(x=125+100=225\) and \(y=2(125)+25=275\). The sum \(x+y=225+275=500\).