AMC 10 · 2014 · #7
Grade 7 arithmeticSuppose A>B>0 and A is x% greater than B. What is x?
Pick an answer.
AMC 10 2014 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: Two positive numbers satisfy $A>B>0$, and $A$ is $x\%$ greater than $B$. Write $x$ in terms of $A$ and $B$, and match it to one of the given formulas.
Givens: $A$ and $B$ are numbers with $A>B>0$; $A$ is $x\%$ greater than $B$; Answer choices are formulas: (A) $100\left(\frac{A-B}{B}\right)$, (B) $100\left(\frac{A+B}{B}\right)$, (C) $100\left(\frac{A+B}{A}\right)$, (D) $100\left(\frac{A-B}{A}\right)$, (E) $100\left(\frac{A}{B}\right)$
Unknowns: The percent $x$, written using $A$ and $B$
Understand
Restated: Two positive numbers satisfy $A>B>0$, and $A$ is $x\%$ greater than $B$. Write $x$ in terms of $A$ and $B$, and match it to one of the given formulas.
Givens: $A$ and $B$ are numbers with $A>B>0$; $A$ is $x\%$ greater than $B$; Answer choices are formulas: (A) $100\left(\frac{A-B}{B}\right)$, (B) $100\left(\frac{A+B}{B}\right)$, (C) $100\left(\frac{A+B}{A}\right)$, (D) $100\left(\frac{A-B}{A}\right)$, (E) $100\left(\frac{A}{B}\right)$
Plan
Primary tool: #13 Convert to Algebra
Secondary: #4 Introduce a Variable, #6 Guess and Check
The phrase "$A$ is $x\%$ greater than $B$" is a word statement hiding one equation. Tool #4 (Introduce a Variable) is already done for us — $x$ is named — so the work is Tool #13 (Convert to Algebra): turn the words into $A = B + \frac{x}{100}B$ and solve for $x$. Because the choices are formulas, Tool #6 (Guess and Check) with concrete numbers gives an independent confirmation that the algebra landed on the right choice.
Execute — Answer: A
6.EE.B.6 Step 1 Translate the percent phrase
- "$A$ is $x\%$ greater than $B$" means you start with $B$ and add $x\%$ of $B$ to it.
- A percent of $B$ is that percent over $100$ times $B$, so $x\%$ of $B$ is $\frac{x}{100}B$.
- Adding it to $B$ gives $A$.
💡 "Greater by a percent" always means the original amount plus that slice of the original amount.
7.RP.A.3 Step 2 Isolate the percent slice
- Subtract $B$ from both sides so the only thing left on one side is the percent slice.
- The left side becomes $A-B$, the difference between the two numbers, and it equals $\frac{x}{100}B$.
💡 The gap $A-B$ is exactly the extra part that the percent increase added on.
6.EE.B.7 Step 3 Solve for $x$
- Divide both sides by $B$ to free the fraction $\frac{x}{100}$, then multiply both sides by $100$ to get $x$ alone.
- This gives $x = 100\left(\frac{A-B}{B}\right)$: the gap $A-B$ compared against the original $B$, scaled to a percent.
- That is choice $\textbf{(A)}$.
💡 Percent increase is always the change divided by the starting value, then times $100$.
6.EE.B.6 "$A$ is $x\%$ greater than $B$" means you start with $B$ and add $x\%$ of $B$ to 7.RP.A.3 Subtract $B$ from both sides so the only thing left on one side is the percent s 6.EE.B.7 Divide both sides by $B$ to free the fraction $\frac{x}{100}$, then multiply bot Review
Reasonableness: The result compares the change $A-B$ to the original $B$, which is what "greater than $B$" demands — the baseline is $B$, so $B$ belongs in the denominator. Since $A>B$, the numerator $A-B$ is positive, so $x>0$, matching "greater." Choices that use $A$ in the denominator (D, E, C) or add instead of subtract (B, C) measure against the wrong baseline or the wrong quantity, so they fail.
Alternative: Guess and Check with numbers: let $B=100$ and $A=110$. Then $A$ is plainly $10\%$ greater than $B$, so $x=10$. Testing choice (A): $100\left(\frac{110-100}{100}\right)=100\cdot\frac{10}{100}=10$. Only (A) returns $10$, confirming the algebra.
CCSS standards used (min grade 7)
6.EE.B.6Use variables to represent numbers and write expressions to solve problems (Turning "$A$ is $x\%$ greater than $B$" into the equation $A = B + \frac{x}{100}B$.)7.RP.A.3Use proportional relationships to solve multi-step ratio and percent problems (Reading the percent increase as the difference $A-B$ measured against the base $B$.)6.EE.B.7Solve real-world problems by writing and solving equations of the form px = q (Isolating $x$ from $A-B=\frac{x}{100}B$ to get $x = 100\left(\frac{A-B}{B}\right)$.)
⭐ Percent increase is the change divided by what you started with, times $100$ — so it is $100$ times $(A-B)$ over $B$.
⭐ Percent increase is the change divided by what you started with, times $100$ — so it is $100$ times $(A-B)$ over $B$.
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