AMC 10 · 2012 · #6

Grade 7 arithmetic

Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In

estimationinterval-arithmetic convert-to-algebra ↑ Prerequisites: interval-arithmetic

📏 Short solution 💡 2 insights

Problem

In order to estimate the value of $x-y$ where $x$ and $y$ are real numbers with $x > y > 0$ , Xiaoli rounded $x$ up by a small amount, rounded $y$ down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct?

Pick an answer.

(A)

$\text{Her estimate is larger than } x-y$

(B)

$\text{Her estimate is smaller than } x-y$

(C)

$\text{Her estimate equals } x-y$

(D)

$\text{Her estimate equals } y-x$

(E)

$\text{Her estimate is } 0$

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.

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Toolkit + CCSS Solution

Understand

Restated: To estimate $x-y$, where $x>y>0$, Xiaoli makes $x$ a little bigger by rounding it up, makes $y$ a little smaller by rounding it down using the same small amount, and then subtracts her two rounded numbers. Decide which statement about her estimate must always be true.

Givens: $x$ and $y$ are real numbers with $x>y>0$; $x$ is rounded up by a small positive amount; $y$ is rounded down by that same amount; Her estimate is the rounded $x$ minus the rounded $y$; Answer choices: (A) estimate is larger than $x-y$, (B) smaller than $x-y$, (C) equals $x-y$, (D) equals $y-x$, (E) is $0$

Unknowns: How her estimate compares to the true value $x-y$

Understand

Plan

Primary tool: #4 Introduce a Variable

Secondary: #13 Convert to Algebra, #3 Eliminate Possibilities

The problem gives no numbers, so the safe move is Tool #4 (Introduce a Variable): call the small rounding amount $k$ and write the two rounded numbers as $x+k$ and $y-k$. Tool #13 (Convert to Algebra) then turns “rounded $x$ minus rounded $y$” into a single expression that can be simplified and compared to $x-y$. Tool #3 (Eliminate Possibilities) reads that comparison against the five statements and keeps the one that must hold for every allowed $x$ and $y$.

Execute — Answer: A

#4 Introduce a Variable 6.EE.B.6 Step 1

Name the rounding amount

The size of the rounding is never given, so name it.
Let $k$ be the small amount, with $k>0$ because a real rounding actually changes the number.
Rounding $x$ up adds $k$, giving $x+k$.
Rounding $y$ down subtracts the same $k$, giving $y-k$.
One letter now stands for the shared rounding amount in both numbers.

$$k>0,\quad \text{rounded }x=x+k,\quad \text{rounded }y=y-k$$

💡 When a quantity is described but not numbered, give it a letter so you can compute with it.

#13 Convert to Algebra 7.EE.A.1 Step 2

Write and simplify her estimate

Her estimate is the rounded $x$ minus the rounded $y$: $(x+k)-(y-k)$.
Removing the parentheses flips the sign of everything in the second group, so $-(y-k)=-y+k$.
The expression becomes $x+k-y+k$, and the two $k$ terms combine into $2k$, giving $x-y+2k$.
So her estimate is exactly the true value $x-y$ with an extra $2k$ added on.

$$(x+k)-(y-k)=x+k-y+k=(x-y)+2k$$

💡 Subtracting a group that was made smaller adds that shrinkage back, so both roundings pile onto the difference.

#3 Eliminate Possibilities 6.NS.C.7 Step 3

Compare to the true value and choose

Compare the estimate $(x-y)+2k$ with the true value $x-y$.
They differ only by $2k$, and since $k>0$, the amount $2k$ is positive.
Adding a positive number always makes a value bigger, so $(x-y)+2k>x-y$ no matter what $x$ and $y$ are.
Her estimate is therefore always larger than $x-y$, which rules out (B), (C), (D), and (E) and leaves choice (A).

$$2k>0\;\Rightarrow\;(x-y)+2k>x-y\;\Rightarrow\;\textbf{(A)}$$

💡 A number plus a positive amount sits to the right of it on the number line, so it is larger.

[1] #4 6.EE.B.6 The size of the rounding is never given, so name it. Let $k$ be the small amount

[2] #13 7.EE.A.1 Her estimate is the rounded $x$ minus the rounded $y$: $(x+k)-(y-k)$. Removing t

[3] #3 6.NS.C.7 Compare the estimate $(x-y)+2k$ with the true value $x-y$. They differ only by $

Review

Reasonableness: Try concrete numbers: let $x=10$, $y=3$, and round by $k=1$. The true difference is $10-3=7$. Xiaoli computes $(10+1)-(3-1)=11-2=9$, which is larger than $7$ by $2$, exactly $2k$. Any other choice of $k>0$ gives the same pattern: the estimate beats the true value by $2k$. This matches the algebra and confirms the estimate is always too high, so (A) is consistent.

Alternative: Skip the variable and reason about how a difference moves. Rounding $x$ up makes the number you start from larger, which raises the difference. Rounding $y$ down means you subtract a smaller number, which also raises the difference. Both changes push the result the same way — upward — so the estimate must land above $x-y$, giving (A) without any equation.

CCSS standards used (min grade 7)

6.EE.B.6 Use variables to represent numbers and write expressions when solving a problem (Naming the unstated rounding amount $k$ and writing the rounded values as $x+k$ and $y-k$.)
7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions with rational coefficients (Expanding $(x+k)-(y-k)$ and combining like terms into $(x-y)+2k$.)
6.NS.C.7 Understand ordering and absolute value of rational numbers (Concluding that adding the positive amount $2k$ makes $(x-y)+2k$ greater than $x-y$.)

⭐ Making the front number bigger and the number you subtract smaller both push a difference up, so this estimate is always too high.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Name the rounding amount

Write and simplify her estimate

Compare to the true value and choose

Review

CCSS standards used (min grade 7)

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