Difference between revisions of "2015 AMC 8 Problems/Problem 7"

Latest revision as of 15:54, 6 January 2024

Problem

Each of two boxes contains three chips numbered $1$ , $2$ , $3$ . A chip is drawn randomly from each box and the numbers on the two chips are multiplied. What is the probability that their product is even?

$\textbf{(A) }\frac{1}{9}\qquad\textbf{(B) }\frac{2}{9}\qquad\textbf{(C) }\frac{4}{9}\qquad\textbf{(D) }\frac{1}{2}\qquad \textbf{(E) }\frac{5}{9}$

Solutions

Solution 1.1

(This solution is similar to Solution 2.) Let's make this a problem with boxes.

In total, there are 9 products derived from these numbers (because 3 numbers per box). This would be our denominator.

Every time a spinner lands on 2, we get an even product. These are $(1,2)$ , $(2,1)$ , $(2,2)$ , $(2,3)$ , and finally $(3,2)$ .

Going back, we see there are $3\cdot3=9$ possible combinations, and we have $5$ evens, the final answer is

$\boxed{\textbf{(E) }\frac{5}{9}}$ .

Solution 1

You can also make this problem into a spinner problem. You have the first spinner with $3$ equally divided

sections: $1, 2$ , and $3$ . You make a second spinner that is identical to the first, with $3$ equal sections of

$1$ , $2$ , and $3$ . If the first spinner lands on $1$ , it must land on two for the result to be even. You write down the first

combination of numbers: $(1,2)$ . Next, if the spinner lands on $2$ , it can land on any number on the second

spinner. We now have the combinations of $(1,2) ,(2,1), (2,2),$ and $(2,3)$ . Finally, if the first spinner ends on $3$ , we

have $(3,2).$ Since there are $3\cdot3=9$ possible combinations, and we have $5$ evens, the final answer is

$\boxed{\textbf{(E) }\frac{5}{9}}$ .

Solution 3

We can also list out the numbers. Box A has chips $1$ , $2$ , and $3$ , and Box B also has chips $1$ , $2$ , and $3$ . Chip $1$ (from Box A)

could be with 3 partners from Box B. This is also the same for chips $2$ and $3$ from Box A. $3+3+3=9$ total sums. Chip $1$ could be multiplied with 2 other chips to make an even product, just like chip $3$ . Chip $2$ can only multiply with 1 chip. $2+2+1=5$ . The answer is $\boxed{\textbf{(E) }\frac{5}{9}}$ .

Solution 4

Here is another way:

Let's start by finding the denominator: Total choices. There are $3$ chips we can choose from in the 1st box, and $3$ chips we can choose from in the 2nd box. We do $3*3$ , and get $9$ . Now - to find the numerator: Desired choices. To get an even number, we need to pick 2 from at least one of the boxes. There are $2$ choices as to finding which box we will draw the 2 from. Then, we have $3$ choices from the other box to pick any of the other chips, $1, 2,$ and $3$ .

$\frac{3 \cdot2}{9} = \frac{6}{9}$

However, we are over counting the $(2,2)$ configuration twice, and so, we subtract that one configuration from our total. $\frac{6}{9} - \frac{1}{9}$ . Thus, our answer is $\boxed{\textbf{(E) }\frac{5}{9}}$ .

~ del-math.

Solution 5

This might take longer to solve, but you definitely will have the right answer. You first list out all the possible combinations regardless if the product is even or odd. $1 \cdot 1 = 1.$ $1 \cdot 2 = 2.$ $1 \cdot 3 = 3.$ $2 \cdot 1 = 2.$ $2 \cdot 2 = 4.$ $2 \cdot 3 = 6.$ $3 \cdot 1 = 3.$ $3 \cdot 2 = 6.$ $3 \cdot 3 = 9.$ There are 9 possible combinations, and $5$ of the combinations’ products are even. So, our answer is $\boxed{\textbf{(E) }\frac{5}{9}}$ .

~MiracleMaths

Video Solution (HOW TO THINK CRITICALLY!!!)

https://youtu.be/oODaaTOHemg

~Education, the Study of Everything

Video Solution

https://youtu.be/PBBrT9iVCVU

~savannahsolver

@@ Line 1: / Line 1: @@
 ==Problem==
 Each of two boxes contains three chips numbered <math>1</math>, <math>2</math>, <math>3</math>. A chip is drawn randomly from each box and the numbers on the two chips are multiplied. What is the probability that their product is even?
 <math>\textbf{(A) }\frac{1}{9}\qquad\textbf{(B) }\frac{2}{9}\qquad\textbf{(C) }\frac{4}{9}\qquad\textbf{(D) }\frac{1}{2}\qquad \textbf{(E) }\frac{5}{9}</math>
 ==Solutions==
-===Solution 1===
+===Solution 1.1===
-Hi! Let's begin. (This solution is similar to Solution 2.)
+(This solution is similar to Solution 2.)
 Let's make this a problem with boxes.
-In total, there are 9 products derived from these numbers (because 3 slips per box). This would be our denominator.
+In total, there are 9 products derived from these numbers (because 3 numbers per box). This would be our denominator.
+Every time a spinner lands on 2, we get an even product. These are <math>(1,2)</math>, <math>(2,1)</math>, <math>(2,2)</math>, <math>(2,3)</math>, and finally <math>(3,2)</math>.
-Every time a spinner lands on 2, we get an even product. These are (1,2), (2,1), (2,2), (2,3), and finally (3,2).
 Going back, we see there are <math>3\cdot3=9</math> possible combinations, and we have <math>5</math> evens, the final answer is
 <math>\boxed{\textbf{(E) }\frac{5}{9}}</math>.
-===Solution 2===
+===Solution 1===
 You can also make this problem into a spinner problem. You have the first spinner with <math>3</math> equally divided

2015 AMC 8 (Problems • Answer Key • Resources)
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