What is the ratio whose terms differ by 40 and the measure of which is $\frac{2}{7}$?

### Concept & Ratio Proportions A simplified ratio $\frac{a}{b}$ can be expressed as actual numerical values $ak$ and $bk$, where $k$ is a common multiplier (or scaling factor). The difference between the actual terms can be represented as the difference between these algebraic expressions. $$ \text{Difference} = |bk - ak| $$ ### Step-by-Step Solution 1. The measure of the ratio is given as $\frac{2}{7}$. Let the actual terms of the ratio be $2k$ and $7k$. 2. The problem states that the terms differ by $40$. We can set up an equation representing this difference: $$ 7k - 2k = 40 $$ 3. Simplify the left side of the equation: $$ 5k = 40 $$ 4. Solve for the common multiplier $k$: $$ k = \frac{40}{5} = 8 $$ 5. Now substitute $k$ back into our expressions to find the actual terms: * First term = $2k = 2 \times 8 = 16$ * Second term = $7k = 7 \times 8 = 56$ 6. The requested ratio is formed by these actual terms, which is $16 : 56$. ### Exam Strategy & Shortcut Look at the difference in the given ratio parts: $7 - 2 = 5$ parts. We know these $5$ parts represent an actual difference of $40$. Therefore, $1$ ratio part = $\frac{40}{5} = 8$. To find the terms, just multiply the original ratio numbers by this factor: $2 \times 8 = 16$ and $7 \times 8 = 56$. The ratio is $16 : 56$. ### Common Pitfall A potential mistake is automatically reducing the final answer back to its simplest form ($2 : 7$). However, the question specifically asks for the unreduced ratio "whose terms differ by 40," meaning you must select the option that shows the exact scaled numbers ($16 : 56$). ### Final Answer Therefore, the correct answer is **16 : 56**.