Difference between revisions of "2021 AMC 12A Problems/Problem 3"

Revision as of 03:54, 14 February 2021

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

Solution 3 (Vertical Addition and Logic)

Let the larger number be $\overline{AB,CD0}.$ It follows that the smaller number is $\overline{A,BCD}.$ Adding vertically, we have $\begin{array}{cccccc} & A & B & C & D & 0 \\ + & & A & B & C & D \\ \hline & 1 & 7 & 4 & 0 & 2 \\ \end{array}$ Working from right to left, we have $D=2\Rightarrow C=8 \Rightarrow B=5 \Rightarrow A=1.$ 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 Solution by Aaron He

https://www.youtube.com/watch?v=xTGDKBthWsw&t=1m28s

Video Solution by Punxsutawney Phil

https://youtube.com/watch?v=MUHja8TpKGw&t=143s

Video Solution by Hawk Math

https://www.youtube.com/watch?v=P5al76DxyHY

Video Solution (Using Algebra and Meta-solving)

https://youtu.be/d2musztzDjw

-pi_is_3.14

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.

@@ Line 22: / Line 22: @@
 ~CoolJupiter 2021
+==Solution 3 (Vertical Addition and Logic)==
+Let the larger number be <math>\overline{AB,CD0}.</math> It follows that the smaller number is <math>\overline{A,BCD}.</math> Adding vertically, we have
+<cmath>\begin{array}{cccccc}
+  & A & B & C & D & 0 \\
++ &   & A & B & C & D \\
+\hline
+  & 1 & 7 & 4 & 0 & 2 \\
+\end{array}</cmath>
+Working from right to left, we have <cmath>D=2\Rightarrow C=8 \Rightarrow B=5 \Rightarrow A=1.</cmath>
+The larger number is <math>15,820</math> and the smaller number is <math>1,582.</math> Their difference is <math>15,820-1,582=\boxed{\textbf{(D)} ~14{,}238}.</math>
+~MRENTHUSIASM
 ==Video Solution by Aaron He==

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