Difference between revisions of "2021 AMC 12A Problems/Problem 5"
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− | {{duplicate|[[2021 AMC 10A Problems | + | {{duplicate|[[2021 AMC 10A Problems/Problem 8|2021 AMC 10A #8]] and [[2021 AMC 12A Problems/Problem 5|2021 AMC 12A #5]]}} |
==Problem== | ==Problem== | ||
− | When a student multiplied the number <math>66</math> by the repeating decimal <cmath>\underline{1}.\underline{a}\underline{b}\underline{a}\underline{b} | + | When a student multiplied the number <math>66</math> by the repeating decimal, |
+ | <cmath>\underline{1}.\underline{a} \ \underline{b} \ \underline{a} \ \underline{b}\ldots=\underline{1}.\overline{\underline{a} \ \underline{b}},</cmath> | ||
+ | where <math>a</math> and <math>b</math> are digits, he did not notice the notation and just multiplied <math>66</math> times <math>\underline{1}.\underline{a} \ \underline{b}.</math> Later he found that his answer is <math>0.5</math> less than the correct answer. What is the <math>2</math>-digit number <math>\underline{a} \ \underline{b}?</math> | ||
<math>\textbf{(A) }15 \qquad \textbf{(B) }30 \qquad \textbf{(C) }45 \qquad \textbf{(D) }60 \qquad \textbf{(E) }75</math> | <math>\textbf{(A) }15 \qquad \textbf{(B) }30 \qquad \textbf{(C) }45 \qquad \textbf{(D) }60 \qquad \textbf{(E) }75</math> | ||
− | ==Solution== | + | ==Solution 1== |
− | It is known that <math>0.\overline{ | + | We are given that <math>66\Bigl(\underline{1}.\overline{\underline{a} \ \underline{b}}\Bigr)-0.5=66\Bigl(\underline{1}.\underline{a} \ \underline{b}\Bigr),</math> from which |
+ | <cmath>\begin{align*} | ||
+ | 66\Bigl(\underline{1}.\overline{\underline{a} \ \underline{b}}\Bigr)-66\Bigl(\underline{1}.\underline{a} \ \underline{b}\Bigr)&=0.5 \\ | ||
+ | 66\Bigl(\underline{1}.\overline{\underline{a} \ \underline{b}} - \underline{1}.\underline{a} \ \underline{b}\Bigr)&=0.5 \\ | ||
+ | 66\Bigl(\underline{0}.\underline{0} \ \underline{0} \ \overline{\underline{a} \ \underline{b}}\Bigr)&=0.5 \\ | ||
+ | 66\left(\frac{1}{100}\cdot\underline{0}.\overline{\underline{a} \ \underline{b}}\right)&=\frac12 \\ | ||
+ | \underline{0}.\overline{\underline{a} \ \underline{b}}&=\frac{25}{33} \\ | ||
+ | \underline{0}.\overline{\underline{a} \ \underline{b}}&=0.\overline{75} \\ | ||
+ | \underline{a} \ \underline{b}&=\boxed{\textbf{(E) }75}. | ||
+ | \end{align*}</cmath> | ||
+ | ~MRENTHUSIASM | ||
+ | |||
+ | ==Solution 2== | ||
+ | It is known that <math>\underline{0}.\overline{\underline{a} \ \underline{b}}=\frac{\underline{a} \ \underline{b}}{99}</math> and <math>\underline{0}.\underline{a} \ \underline{b}=\frac{\underline{a} \ \underline{b}}{100}.</math> | ||
+ | |||
+ | Let <math>x=\underline{a} \ \underline{b}.</math> We have <cmath>66\biggl(1+\frac{x}{99}\biggr)-66\biggl(1+\frac{x}{100}\biggr)=0.5.</cmath> Expanding and simplifying give <math>\frac{x}{150}=0.5,</math> so <math>x=\boxed{\textbf{(E) }75}.</math> | ||
+ | |||
+ | ~aop2014 ~BakedPotato66 ~MRENTHUSIASM | ||
+ | |||
+ | ==Solution 3 (Similar to Solution 2)== | ||
+ | |||
+ | We have <cmath>66 \cdot \left(1 + \frac{10a+b}{100}\right) + \frac{1}{2} = 66 \cdot \left(1+ \frac{10a+b}{99}\right).</cmath> | ||
+ | Expanding both sides, we have <cmath>66 + \frac{33(10a+b)}{50} + \frac{1}{2} = 66 + \frac{2(10a+b)}{3}.</cmath> | ||
+ | Subtracting <math>66</math> from both sides, we have <cmath>\frac{33(10a+b)}{50} + \frac{1}{2} = \frac{2(10a+b)}{3}.</cmath> | ||
+ | Multiplying both sides by <math>50 \cdot 3 = 150,</math> we have <cmath>99(10a+b) + 75 = 100(10a+b).</cmath> | ||
+ | Thus, the answer is <math>10a+b = \boxed{\textbf{(E) }75}.</math> | ||
+ | |||
+ | By letting <math>x=\underline{a} \ \underline{b}=10a+b,</math> this solution is similar to Solution 2. In this solution, we solve for <math>10a+b</math> as a whole. | ||
+ | |||
+ | -mathboy282 (Solution) | ||
+ | |||
+ | ~MRENTHUSIASM (Minor Revision) | ||
+ | |||
+ | ==Solution 4 (Answer Choices & Modular) == | ||
+ | |||
+ | Let <math>\underline{a} \ \underline{b}</math> represent the two-digit number <math>ab</math>, not the product of the digits <math>a</math> and <math>b</math>. We can construct fractions for the values <math>1.\underline{a} \ \underline{b}</math>, and <math>1.\overline{\underline{a} \ \underline{b}}</math>, which are <math>\frac{1\underline{a} \ \underline{b}}{100}</math> and <math>\frac{1\underline{a} \ \underline{b}}{99}</math> respectively. Multiplying by <math>66</math> on both sides and adding <math>1/2</math> to <math>\frac{1\underline{a} \ \underline{b}}{100}</math> and simplifying, we get this: <cmath>\frac{33 * \underline{a}\underline{b} + 25}{50} = \frac{2\underline{a}\underline{b}}{3}.</cmath> | ||
+ | |||
+ | Looking at the answer choices, we notice that all of them are divisible by <math>3</math>. This means that since the right-hand side will result in an integer, the left-hand side needs to as well. This means that the numerator of the left-hand side fraction has to be divisible by <math>50</math>. So, we get this expression: | ||
+ | |||
+ | <math>33 * \underline{a}\underline{b}\ + 25 \equiv 0\pmod{50}. </math> | ||
+ | |||
+ | <math>33 * \underline{a}\underline{b}\ \equiv -25\pmod{50}. </math> | ||
+ | |||
+ | <math>33 * \underline{a}\underline{b}\ \equiv 25\pmod{50}. </math> | ||
+ | |||
+ | This means that the product of <math>33</math> and <math>\underline{a}\underline{b}</math> must have a remainder of <math>25</math> when divided by <math>50</math>. Since it must have a remainder of <math>25</math>, the product should have a units digit of <math>5</math>, which eliminates <math>\textbf{(B) }</math> and <math>\textbf{(D) }</math>. Multiplying <math>33</math> to the rest of the answer choices, the only one which fills this requirement is <math>75</math>, which is <math>\boxed{\textbf{(E) }75}.</math> | ||
+ | |||
+ | ~neeyakkid23 | ||
+ | |||
+ | ==Video Solution (Simple & Quick)== | ||
+ | https://youtu.be/9HI79V-vtCU | ||
+ | |||
+ | ~ Education, the Study of Everything | ||
==Video Solution by Aaron He== | ==Video Solution by Aaron He== | ||
https://www.youtube.com/watch?v=xTGDKBthWsw&t=4m12s | https://www.youtube.com/watch?v=xTGDKBthWsw&t=4m12s | ||
− | ==Video Solution(Use of | + | ==Video Solution (Use of Properties of Repeating Decimals) == |
https://www.youtube.com/watch?v=zS1u-ohUDzQ&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=6\ | https://www.youtube.com/watch?v=zS1u-ohUDzQ&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=6\ | ||
Line 23: | Line 77: | ||
https://www.youtube.com/watch?v=P5al76DxyHY | https://www.youtube.com/watch?v=P5al76DxyHY | ||
− | == Video Solution (Using | + | == Video Solution by OmegaLearn (Using Repeating Decimal Properties) == |
https://youtu.be/vQZ13WiL4WU | https://youtu.be/vQZ13WiL4WU | ||
~ pi_is_3.14 | ~ pi_is_3.14 | ||
− | ==Video Solution | + | ==Video Solution== |
− | https://youtu.be/ | + | https://youtu.be/DOF3FYUsXsU |
+ | |||
+ | ~savannahsolver | ||
+ | |||
+ | ==Video Solution by TheBeautyofMath== | ||
+ | https://youtu.be/s6E4E06XhPU?t=360 (AMC 10A) | ||
+ | |||
+ | https://youtu.be/rEWS75W0Q54?t=511 (AMC 12A) | ||
+ | |||
+ | ~IceMatrix | ||
− | + | ==Video Solution by The Learning Royal== | |
+ | https://youtu.be/AWjOeBFyeb4 | ||
==See also== | ==See also== |
Latest revision as of 10:18, 28 September 2024
- The following problem is from both the 2021 AMC 10A #8 and 2021 AMC 12A #5, so both problems redirect to this page.
Contents
- 1 Problem
- 2 Solution 1
- 3 Solution 2
- 4 Solution 3 (Similar to Solution 2)
- 5 Solution 4 (Answer Choices & Modular)
- 6 Video Solution (Simple & Quick)
- 7 Video Solution by Aaron He
- 8 Video Solution (Use of Properties of Repeating Decimals)
- 9 Video Solution by Punxsutawney Phil
- 10 Video Solution by Hawk Math
- 11 Video Solution by OmegaLearn (Using Repeating Decimal Properties)
- 12 Video Solution
- 13 Video Solution by TheBeautyofMath
- 14 Video Solution by The Learning Royal
- 15 See also
Problem
When a student multiplied the number by the repeating decimal, where and are digits, he did not notice the notation and just multiplied times Later he found that his answer is less than the correct answer. What is the -digit number
Solution 1
We are given that from which ~MRENTHUSIASM
Solution 2
It is known that and
Let We have Expanding and simplifying give so
~aop2014 ~BakedPotato66 ~MRENTHUSIASM
Solution 3 (Similar to Solution 2)
We have Expanding both sides, we have Subtracting from both sides, we have Multiplying both sides by we have Thus, the answer is
By letting this solution is similar to Solution 2. In this solution, we solve for as a whole.
-mathboy282 (Solution)
~MRENTHUSIASM (Minor Revision)
Solution 4 (Answer Choices & Modular)
Let represent the two-digit number , not the product of the digits and . We can construct fractions for the values , and , which are and respectively. Multiplying by on both sides and adding to and simplifying, we get this:
Looking at the answer choices, we notice that all of them are divisible by . This means that since the right-hand side will result in an integer, the left-hand side needs to as well. This means that the numerator of the left-hand side fraction has to be divisible by . So, we get this expression:
This means that the product of and must have a remainder of when divided by . Since it must have a remainder of , the product should have a units digit of , which eliminates and . Multiplying to the rest of the answer choices, the only one which fills this requirement is , which is
~neeyakkid23
Video Solution (Simple & Quick)
~ Education, the Study of Everything
Video Solution by Aaron He
https://www.youtube.com/watch?v=xTGDKBthWsw&t=4m12s
Video Solution (Use of Properties of Repeating Decimals)
https://www.youtube.com/watch?v=zS1u-ohUDzQ&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=6\
~North America Math Contest Go Go Go
Video Solution by Punxsutawney Phil
https://youtube.com/watch?v=MUHja8TpKGw&t=359s
Video Solution by Hawk Math
https://www.youtube.com/watch?v=P5al76DxyHY
Video Solution by OmegaLearn (Using Repeating Decimal Properties)
~ pi_is_3.14
Video Solution
~savannahsolver
Video Solution by TheBeautyofMath
https://youtu.be/s6E4E06XhPU?t=360 (AMC 10A)
https://youtu.be/rEWS75W0Q54?t=511 (AMC 12A)
~IceMatrix
Video Solution by The Learning Royal
See also
2021 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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 4 |
Followed by Problem 6 |
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.