If $40 : 35 :: 35 : k$, find the value of $k - 1$. | Mathematics MCQ | Examoogle

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If $40: 35:: 35: k$ , find the value of $k - 1$ .

$\frac{49}{16}$

$\frac{245}{8}$

$\frac{49}{8}$

$\frac{49}{16}$

Correct Answer

📐 MA • Math Direct FormulaMathematicsRatio & Proportion

Option B

$\frac{245}{8}$

Quick Summary:

Use the property of mean proportionals: $k = \frac{b^2}{a}$ . Calculate $\frac{35^2}{40} = \frac{1225}{40} = \frac{245}{8}$ . Then find $k-1 = \frac{245}{8} - 1 = \frac{237}{8}$ .

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📐MAMath SolutionDirect Formula

📋 Given

The proportion 40 : 35 :: 35 : k.

🔢 Formula Used

ab=cd  ⟹  a×d=b×c\frac{a}{b} = \frac{c}{d} \implies a \times d = b \times cba​=dc​⟹a×d=b×c

⚡ Exam Hall Shortcut / Speed Trick

Use the property of mean proportionals: $k = \frac{b^2}{a}$ . Calculate $\frac{35^2}{40} = \frac{1225}{40} = \frac{245}{8}$ . Then find $k-1 = \frac{245}{8} - 1 = \frac{237}{8}$ .

⚠️ Common Student Trap / Pitfall

Many students calculate the value of $k$ correctly as $\frac{245}{8}$ but forget to subtract $1$ as asked in the question.

🔢 Step-by-Step Solution

Establish the proportionality

Given the proportion $40: 35:: 35: k$ , we write it as an equation of ratios: $\frac{40}{35} = \frac{35}{k}$ .

4035=35k\frac{40}{35} = \frac{35}{k}3540​=k35​

Solve for k

Cross-multiplying, we get $40k = 35 \times 35$ . Calculating the product $35 \times 35 = 1225$ , so $40k = 1225$ . Dividing both sides by 40: $k = \frac{1225}{40}$ . Simplifying by dividing both numerator and denominator by 5, we get $k = \frac{245}{8}$ .

k=122540=2458k = \frac{1225}{40} = \frac{245}{8}k=401225​=8245​

Find k - 1

Now we compute $k - 1$ . Substituting $k = \frac{245}{8}$ , we get $\frac{245}{8} - 1 = \frac{245}{8} - \frac{8}{8} = \frac{237}{8}$ .

k−1=2458−1=2378k - 1 = \frac{245}{8} - 1 = \frac{237}{8}k−1=8245​−1=8237​

✅

A is correct because the value of k is $\frac{245}{8}$ , and subtracting 1 from it yields $\frac{237}{8}$ .

Core Concepts Used

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Proportion Mean Proportional Fractional Subtraction

💡 EXAM TIP

This concept of mean proportionals is widely used in geometry for calculating geometric means and in speed/time problems involving average rates.

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