Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of $27.50 for dinner. What is the cost of her dinner without tax or tip?
The sales tax of 10% combined with the tip of 15% add 25% to the bill. So each dollar the meal cost will contribute $1.25 to the bill. Below is a table showing different costs of the meal along with the total price including sales tax and tip:
| Meal Cost (in dollars) | Total cost with tax and tip (in dollars) |
| 1 | 1.25 |
| 5 | 6.25 |
| 10 | 12.50 |
| 15 | 18.75 |
| 20 | 25 |
| 25 | 31.25 |
This shows that the cost of the meal is between $20 and $25 so we can try $1 increments here:
| Meal Cost (in dollars) | Total cost with tax and tip (in dollars) |
| 21 | 26.25 |
| 22 | 27.50 |
This shows that the cost of the meal was $22.
Notice that the method employed here will require patience if the cost of the meal is not an even number of dollars. First we would find which two whole dollar amounts it is between and then would have to start a new table with increments in ten cents and, eventually, cents if necessary.
Anna paid $27.50 for the meal, the tax, and the tip. The tax is 10% of the price of the meal and the tip is 15% of the price of the meal. Combining the meal, the tax, and the tip we get 125% of the cost of the meal. Since Anna paid $27.50 total this means that 1.25 times the cost of the meal is $27.50. So dividing $27.50 by 1.25 gives the cost of the meal: $$ \frac{$ 27.50}{1.25} = $22. $$
If \(x\) equals the cost of the meal without tax and tip, then the tax is 10 hundredths that amount:
$$\frac{10}{100}\cdot x = 0.10x$$
and the tip is 15 hundredths that amount:
$$\frac{15}{100}\cdot x =0.15x$$
Using the distributive property, we can see that the total cost for the meal with tax and tip is
$$x+ 0.10x+0.15x=(1+0.10+0.15)x = 1.25x$$
Since the total cost of the meal is $27.50, we know that is
$$1.25x=27.50$$ and that $$x=\frac{27.50}{1.25}=22$$
So the cost of the meal without tax and tip is $22.
Commentary
The purpose of this task is to give students an opportunity to solve a multi-step percentage problem that can be approached in many ways. Because the tax and tip in this problem are a fixed percentage of the cost of the meal, the total amount paid is in a fixed ratio with the cost of the meal. The task can illustrate multiple standards depending on the tools students have already developed and the way the approach the problem. Some students might use a ratio table; others might divide by the appropriate unit rate. It can also be solved by setting up an equation with \(x\) representing the cost of the meal. Such an algebraic approach is challenging but within reach of 6th graders as the third solution shows. However, it would be best to give it to 6th graders in an instructional setting as problem-solving exercise because sixth graders will still be working to integrate these different ideas.
This task was adapted from problem #5 on the 2012 American Mathematics Competition (AMC) 10B Test. The responses to the multiple choice answers for the problem had the following distribution: