2021 AMC 12A Problems/Problem 3

The following problem is from both the 2021 AMC 10A #3 and 2021 AMC 12A #3, so both problems redirect to this page.

Problem

The sum of two natural numbers is $17{,}402$ . One of the two numbers is divisible by $10$ . If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?

$\textbf{(A)} ~10{,}272\qquad\textbf{(B)} ~11{,}700\qquad\textbf{(C)} ~13{,}362\qquad\textbf{(D)} ~14{,}238\qquad\textbf{(E)} ~15{,}426$

Solution 1

The units digit of a multiple of $10$ will always be $0$ . We add a $0$ whenever we multiply by $10$ . So, removing the units digit is equal to dividing by $10$ .

Let the smaller number (the one we get after removing the units digit) be $a$ . This means the bigger number would be $10a$ .

We know the sum is $10a+a = 11a$ so $11a=17402$ . So $a=1582$ . The difference is $10a-a = 9a$ . So, the answer is $9(1582) = 14238 = \boxed{\textbf{(D)}}$ .

--abhinavg0627

Solution 2 (Lazy Speed)

Since the ones place of a multiple of $10$ is $0$ , this implies the other integer has to end with a $2$ since both integers sum up to a number that ends with a $2$ . Thus, the ones place of the difference has to be $10-2=8$ , and the only answer choice that ends with an $8$ is $\boxed{\textbf{(D)}~14238}$

~CoolJupiter 2021

Another quick solution is to realize that the sum is represents a number $n$ added to $10n$ . The difference is $9n$ , which is $\frac{9}{11}$ of the given sum.

Solution 3 (Vertical Addition and Logic)

Let the larger number be $\underline{AB{,}CD0}.$ It follows that the smaller number is $\underline{A{,}BCD}.$ Adding vertically, we have $\begin{array}{cccccc} & A & B & C & D & 0 \\ + & & A & B & C & D \\ \hline & & & & & \\ [-2.5ex] & 1 & 7 & 4 & 0 & 2 \\ \end{array}$ Working from right to left, we get $D=2\implies C=8 \implies B=5 \implies A=1.$ The larger number is $15{,}820$ and the smaller number is $1{,}582.$ Their difference is $15{,}820-1{,}582=\boxed{\textbf{(D)} ~14{,}238}.$

~MRENTHUSIASM

Video Solutions

(Simple)

https://youtu.be/SEp9flDYm2c

~ Education, the study Of Everything

-pi_is_3.14

by WhyMath

https://youtu.be/VpYmQEKcBpA

~savannahsolver

by TheBeautyofMath

https://youtu.be/50CThrk3RcM?t=107 (for AMC 10A)

by IceMatrix

https://youtu.be/rEWS75W0Q54?t=198 (for AMC 12A)

~IceMatrix

(Problems 1-3)

https://youtu.be/CupJpUzKPB0

~MathWithPi

by The Learning Royal

https://youtu.be/slVBYmcDMOI

2021 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

2021 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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2021 AMC 12A Problems/Problem 3

Contents

Problem

Solution 1

Solution 2 (Lazy Speed)

Solution 3 (Vertical Addition and Logic)

Video Solutions

(Simple)

by North America Math Contest Go Go Go

by Aaron He

by Punxsutawney Phil

by Hawk Math

(Using Algebra and Meta-solving)

by WhyMath

by TheBeautyofMath

by IceMatrix

(Problems 1-3)

by The Learning Royal

See also