AMC 10 · 2012 · #5
Grade 7 arithmeticAnna 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 dollars for dinner. What is the cost of her dinner without tax or tip in dollars?
Pick an answer.
AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A meal has a base price. A $15\%$ tip and a $10\%$ sales tax are each figured on that base price, and both are added to it. The three amounts together come to $\$27.50$. Find the base price of the meal.
Givens: The sales tax is $10\%$ of the meal's pre-tip price; The tip is $15\%$ of the meal's price before tax; Both the tip and the tax are computed from the same base price, not from each other; The total paid is $\$27.50$; Answer choices: (A) $18$, (B) $20$, (C) $21$, (D) $22$, (E) $24$
Unknowns: The cost of the dinner with no tax and no tip added
Understand
Restated: A meal has a base price. A $15\%$ tip and a $10\%$ sales tax are each figured on that base price, and both are added to it. The three amounts together come to $\$27.50$. Find the base price of the meal.
Givens: The sales tax is $10\%$ of the meal's pre-tip price; The tip is $15\%$ of the meal's price before tax; Both the tip and the tax are computed from the same base price, not from each other; The total paid is $\$27.50$; Answer choices: (A) $18$, (B) $20$, (C) $21$, (D) $22$, (E) $24$
Plan
Primary tool: #4 Introduce a Variable
Secondary: #13 Convert to Algebra, #3 Eliminate Possibilities
The base price is the one unknown that everything else is built from, so Tool #4 (Introduce a Variable) names it once and writes the tip and tax as $0.15x$ and $0.10x$. Tool #13 (Convert to Algebra) then turns the sentence “meal plus tip plus tax equals $27.50$” into a single equation and collects the like terms into one multiple of $x$. Tool #3 (Eliminate Possibilities) checks the result against the built-in trap answers, such as the value you get if you wrongly tax the tip too.
Execute — Answer: D
6.EE.B.6 Step 1 Name the base price
- Let $x$ be the cost of the dinner before any tax or tip.
- Because both extra charges are figured from this same base price, the tip is $15\%$ of $x$, which is $0.15x$, and the tax is $10\%$ of $x$, which is $0.10x$.
- Writing both charges in terms of one letter keeps them tied to the same starting number.
💡 One unknown drives everything, so give it a single name and describe the rest with it.
7.RP.A.3 Step 2 Add the parts into one equation
- The total is the meal plus the tip plus the tax: $x+0.15x+0.10x$.
- These are like terms — all multiples of $x$ — so add the coefficients $1+0.15+0.10=1.25$.
- The total spent is $\$27.50$, giving the equation $1.25x=27.50$. Notice the tip and tax are each taken from $x$, so they are added on, never multiplied together.
💡 Adding a $15\%$ tip and a $10\%$ tax to a price just scales it to $125\%$ of itself.
6.EE.B.7 Step 3 Solve for the base price
- Undo the multiplication by dividing both sides by $1.25$: $x=27.50\div1.25$.
- Since $1.25=\tfrac{5}{4}$, dividing by $1.25$ is the same as multiplying by $\tfrac{4}{5}$, so $x=27.50\times\tfrac{4}{5}=22$.
- The dinner costs $\$22$ before tax and tip.
💡 An equation of the form $px=q$ is solved in one step by dividing the total by $p$.
6.RP.A.3 Step 4 Check the total and pick the answer
- Test $x=22$: the tip is $0.15\times22=3.30$ and the tax is $0.10\times22=2.20$, so the total is $22+3.30+2.20=27.50$, which matches.
- That is choice (D).
- The trap answer $20$ in (B) comes from taxing the tipped amount instead of the base, and the other choices do not hit $27.50$ exactly.
💡 Plugging the answer back into the original words is the surest way to confirm it.
6.EE.B.6 Let $x$ be the cost of the dinner before any tax or tip. Because both extra char 7.RP.A.3 The total is the meal plus the tip plus the tax: $x+0.15x+0.10x$. These are like 6.EE.B.7 Undo the multiplication by dividing both sides by $1.25$: $x=27.50\div1.25$. Sin 6.RP.A.3 Test $x=22$: the tip is $0.15\times22=3.30$ and the tax is $0.10\times22=2.20$, Review
Reasonableness: The base price must be well under the $\$27.50$ total, since a quarter of it is added on as tip and tax. A quick estimate: $25\%$ extra on about $\$22$ is roughly $\$5.50$, and $22+5.50=27.50$ lands exactly on the total, so $\$22$ is consistent. Choice (E) $24$ would already push the total past $27.50$ once the $25\%$ is added, and the smaller choices leave the total short, so only (D) fits.
Alternative: Skip separate terms and reason with one percent. Adding a $15\%$ tip and a $10\%$ tax to a base price turns it into $125\%$ of that price, i.e. $1.25$ times it. So the base price is simply $27.50\div1.25=22$, the same answer in a single division.
CCSS standards used (min grade 7)
6.EE.B.6Use variables to represent numbers and write expressions when solving a problem (Naming the base price $x$ and writing the tip and tax as $0.15x$ and $0.10x$.)7.RP.A.3Use proportional relationships to solve multi-step ratio and percent problems (Combining a $15\%$ tip and $10\%$ tax on the same base into the factor $1.25$ and the equation $1.25x=27.50$.)6.EE.B.7Solve real-world and mathematical problems by writing and solving equations of the form px = q (Solving $1.25x=27.50$ for $x$ by dividing both sides by $1.25$.)6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Verifying $x=22$ by recomputing the $15\%$ tip and $10\%$ tax and checking they sum to $27.50$.)
⭐ When a tip and a tax are both taken from the same price, add their percents to the whole and divide the total by that single factor.
⭐ When a tip and a tax are both taken from the same price, add their percents to the whole and divide the total by that single factor.
More like this
Same archetype — closest grade level first.