Sameer chose 12 different toppings for his frozen yogurt sundae, which was $\frac{3}{4}$ of the total number of different toppings available at the make-your-own sundae shop. To determine the number of different toppings available at the shop, Sameer set up and solved the equation as shown below.
$\begin{aligned}
\frac{3}{4} & =\frac{x}{12} \\
\frac{3}{4}(12) & =\frac{x}{12}(12) \\
9 & =x
\end{aligned}
Which best describes the error that Sameer made?
A. Sameer did not use the correct equation to model the given information.
B. Sameer should have multiplied both sides of the equation by $\frac{4}{3}$ instead of by 12.
Answer (2)
Define T as the total number of toppings.
Set up the correct equation: 4 3 T = 12 .
Solve for T : T = 16 .
Sameer's equation 4 3 = 12 x is incorrect because it doesn't model the given information.
The error is that Sameer did not use the correct equation to model the given information.
Explanation
Setting up the correct equation Let T be the total number of toppings available at the shop. We know that Sameer chose 12 toppings, which represents 4 3 of the total number of toppings. Therefore, we can set up the equation: 4 3 T = 12
Solving for T To solve for T , we need to isolate T on one side of the equation. We can do this by multiplying both sides of the equation by the reciprocal of 4 3 , which is 3 4 : 3 4 × 4 3 T = 3 4 × 12 T = 3 4 × 12 T = 3 48 T = 16 So, the total number of toppings available at the shop is 16.
Analyzing Sameer's equation Now let's analyze Sameer's equation: 4 3 = 12 x This equation implies that 4 3 is equal to the ratio of some number x to 12. However, the problem states that 12 toppings represent 4 3 of the total toppings. Sameer's equation incorrectly sets 4 3 equal to a fraction with 12 in the denominator, which doesn't model the given information correctly.
Identifying the error Sameer's error was setting up the wrong equation. The correct equation should have been 4 3 T = 12 , where T is the total number of toppings. Sameer's equation, 4 3 = 12 x , implies that x is 4 3 of 12, which is 9. This is not what the problem states.
Conclusion The correct answer is that Sameer did not use the correct equation to model the given information.
Examples
Imagine you're baking a cake and the recipe says you need 4 3 of a cup of sugar. If you know that 4 3 of the total amount of sugar you have is equal to 12 tablespoons, you can use the same method to find out how many tablespoons of sugar you have in total. This kind of problem helps you understand proportions and how to find the whole amount when you know a fraction of it.
Sameer's error was in setting up the equation incorrectly, as he should have modeled it with 4 3 T = 12 instead of 4 3 = 12 x . This mistake led him to an incorrect interpretation of the total toppings. Therefore, the correct choice is A.
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