The sum of two numbers is 22. Five times one number is equal to 6 times the other. The bigger of the two numbers is

## Concept & Logic When a multiple of one quantity equals a multiple of another, their ratio is the inverse of those multipliers. This ratio can then be distributed over their total sum. $$ a \cdot x = b \cdot y \Rightarrow \frac{x}{y} = \frac{b}{a} $$ ## Step-by-Step Solution * **Given:** Let the numbers be $x$ and $y$. Sum $x + y = 22$. Also, $5x = 6y$. * **Calculation / Deduction:** * From $5x = 6y$, we find the ratio of $x$ to $y$: * $$ \frac{x}{y} = \frac{6}{5} $$ * This means the numbers are in the ratio of $6:5$. Let them be $6k$ and $5k$. * Substitute into the sum equation: * $6k + 5k = 22$ * $11k = 22$ * $k = 2$ * The numbers are $6(2) = 12$ and $5(2) = 10$. * The bigger number is $12$. ## Exam Strategy & Shortcut **Mental Math via Proportions:** $5$ times one is $6$ times the other, meaning the numbers are in a $6:5$ ratio. Total parts $= 11$. Sum is $22$, which is exactly double the total parts. Therefore, the numbers are just double the ratio figures: $12$ and $10$. The larger is instantly $12$. ## Common Pitfall A frequent mistake is writing the ratio as $5:6$ instead of $6:5$ when translating $5x = 6y$. Always double-check inverse proportionality: the number with the smaller coefficient must be the larger value to balance the equation. ## Final Answer **Therefore, the correct answer is 12.**