AMC 10 · 2012 · #6
Grade 8 arithmeticThe product of two positive numbers is 9. The reciprocal of one of these numbers is 4 times the reciprocal of the other number. What is the sum of the two numbers?
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: Two positive numbers multiply to 9. The reciprocal of one number is 4 times the reciprocal of the other. Find the sum of the two numbers.
Givens: The two numbers are positive.; Their product is 9.; The reciprocal of one number equals 4 times the reciprocal of the other.
Unknowns: The two numbers themselves.; Their sum.
Understand
Restated: Two positive numbers multiply to 9. The reciprocal of one number is 4 times the reciprocal of the other. Find the sum of the two numbers.
Givens: The two numbers are positive.; Their product is 9.; The reciprocal of one number equals 4 times the reciprocal of the other.
Plan
Primary tool: #4 Introduce a Variable
Secondary: #13 Convert to Algebra
Two unknown numbers are tied together by two facts: a product and a reciprocal relationship. Naming each number with a letter lets me turn both facts into equations, then combine them into one equation in a single variable and solve it.
Execute — Answer: D
6.EE.B.6 Step 1 Name the two numbers
- Call the two positive numbers $a$ and $b$.
- Giving them letters lets me write down each fact as an equation instead of juggling words.
💡 A letter is just a placeholder for a number you don't know yet.
6.EE.A.2 Step 2 Turn the two facts into equations
- "The product is 9" becomes $ab = 9$.
- "The reciprocal of one is 4 times the reciprocal of the other" becomes $\frac{1}{a} = 4\cdot\frac{1}{b}$.
💡 Each English sentence maps directly onto one equation.
7.EE.B.4 Step 3 Simplify the reciprocal equation
- In $\frac{1}{a} = \frac{4}{b}$, cross-multiply: $b = 4a$.
- So one number is 4 times the other.
💡 A smaller number has a larger reciprocal, so the one with the bigger reciprocal is the smaller number.
8.EE.A.2 Step 4 Substitute and solve for a
- Replace $b$ with $4a$ in $ab = 9$: $a\cdot 4a = 9$, so $4a^2 = 9$ and $a^2 = \frac{9}{4}$.
- Since $a$ is positive, $a = \frac{3}{2}$.
💡 Substituting collapses two unknowns into one, and taking the positive square root undoes the squaring.
5.NF.A.1 Step 5 Find b and add
- Then $b = 4a = 4\cdot\frac{3}{2} = 6$.
- The sum is $\frac{3}{2} + 6 = \frac{3}{2} + \frac{12}{2} = \frac{15}{2}$.
- So the answer is (D).
💡 Write the whole number as halves so both terms share a denominator before adding.
6.EE.B.6 Call the two positive numbers $a$ and $b$. Giving them letters lets me write dow 6.EE.A.2 "The product is 9" becomes $ab = 9$. "The reciprocal of one is 4 times the recip 7.EE.B.4 In $\frac{1}{a} = \frac{4}{b}$, cross-multiply: $b = 4a$. So one number is 4 tim 8.EE.A.2 Replace $b$ with $4a$ in $ab = 9$: $a\cdot 4a = 9$, so $4a^2 = 9$ and $a^2 = \fr 5.NF.A.1 Then $b = 4a = 4\cdot\frac{3}{2} = 6$. The sum is $\frac{3}{2} + 6 = \frac{3}{2} Review
Reasonableness: Check both facts with $a=\frac{3}{2}$ and $b=6$: the product is $\frac{3}{2}\cdot 6 = 9$ (correct), and $\frac{1}{a} = \frac{2}{3}$ while $\frac{1}{b} = \frac{1}{6}$, and indeed $\frac{2}{3} = 4\cdot\frac{1}{6}$ (correct). The sum $\frac{15}{2} = 7.5$ matches choice (D).
Alternative: Instead of algebra, guess and check: look for two numbers where one is 4 times the other and the product is 9. Testing $\frac{3}{2}$ and $6$ gives $\frac{3}{2}\cdot 6 = 9$ right away, and their sum is $\frac{15}{2}$.
CCSS standards used (min grade 8)
6.EE.B.6Use variables to represent numbers and write expressions to solve problems (Naming the two unknown numbers with letters.)6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers (Translating the product and reciprocal facts into equations.)7.EE.B.4Use variables to represent quantities and construct simple equations and inequalities (Cross-multiplying the reciprocal equation into b = 4a.)8.EE.A.2Use square root and cube root symbols to represent solutions (Solving a^2 = 9/4 by taking the positive square root.)5.NF.A.1Add and subtract fractions with unlike denominators (Adding 3/2 + 6 to get the final sum 15/2.)
⭐ Turn each sentence into an equation, use one fact to replace a letter, and you shrink two unknowns down to one you can solve.
⭐ Turn each sentence into an equation, use one fact to replace a letter, and you shrink two unknowns down to one you can solve.
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