Five mangoes and four oranges cost as much as three mangoes and seven oranges. What is the ratio of the cost of one mango to the cost of one orange?

### Concept & Linear Equations To find the ratio of the costs, we can set up a linear equation based on the given statement. By assigning variables to the cost of each fruit, we can equate the two combinations and solve for the ratio of the two variables. $$ \text{Cost}_1 = \text{Cost}_2 $$ ### Step-by-Step Solution 1. Let the cost of one mango be $m$ and the cost of one orange be $o$. 2. Translate the given statement into an equation: "Five mangoes and four oranges cost as much as three mangoes and seven oranges." $$ 5m + 4o = 3m + 7o $$ 3. Group the like terms by moving all $m$ terms to one side and all $o$ terms to the other side: $$ 5m - 3m = 7o - 4o $$ 4. Simplify the equation: $$ 2m = 3o $$ 5. Rearrange the equation to find the ratio of the cost of one mango to one orange ($\frac{m}{o}$): $$ \frac{m}{o} = \frac{3}{2} $$ 6. This gives the ratio $3 : 2$. ### Exam Strategy & Shortcut You can solve this mentally by looking at the differences. The difference in mangoes ($5 - 3 = 2$) must exactly balance the difference in oranges ($7 - 4 = 3$). Therefore, $2 \text{ Mangoes} = 3 \text{ Oranges}$. Reversing the coefficients gives the ratio $m : o = 3 : 2$. ### Common Pitfall A frequent mistake is to leave the equation as $2m = 3o$ and incorrectly conclude the ratio is $2 : 3$. Remember that to get $m$ over $o$, you divide the coefficient of $o$ by the coefficient of $m$. ### Final Answer Therefore, the correct answer is **3 : 2**.