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Answer :
Sure! Let's go through the steps needed to solve this problem:
1. Identify the problem and assign variables:
- There are a total of 120 counters in a box, consisting only of red and blue counters.
- The red counters are three times the number of blue counters.
2. Set up equations:
- Let's say the number of blue counters is [tex]\( x \)[/tex].
- Then, the number of red counters would be [tex]\( 3x \)[/tex].
- According to the problem, the total number of counters is [tex]\( x + 3x = 4x = 120 \)[/tex].
3. Solve for [tex]\( x \)[/tex]:
- [tex]\( 4x = 120 \)[/tex]
- Divide both sides by 4 to find the number of blue counters:
[tex]\[
x = \frac{120}{4} = 30
\][/tex]
- So, there are 30 blue counters initially.
4. Find the initial number of red counters:
- Since red counters are three times the number of blue counters:
[tex]\[
3x = 3 \times 30 = 90
\][/tex]
- There are 90 red counters initially.
5. Calculate the counters taken:
- Carl takes [tex]\( \frac{1}{3} \)[/tex] of the red counters:
[tex]\[
\frac{1}{3} \times 90 = 30
\][/tex]
So, 30 red counters are taken by Carl.
- Kerry takes [tex]\( 80\% \)[/tex] of the blue counters:
[tex]\[
0.8 \times 30 = 24
\][/tex]
So, 24 blue counters are taken by Kerry.
6. Calculate the counters left:
- Red counters left: [tex]\( 90 - 30 = 60 \)[/tex].
- Blue counters left: [tex]\( 30 - 24 = 6 \)[/tex].
7. Find the ratio of remaining red to blue counters:
- The number of red counters left is 60, and the number of blue counters left is 6.
- The ratio of red counters to blue counters is:
[tex]\[
\frac{60}{6} = 10
\][/tex]
- Therefore, the simplified ratio of the number of red counters to blue counters now in the box is [tex]\( 10:1 \)[/tex].
I hope this explanation helps you understand each step in solving the problem!
1. Identify the problem and assign variables:
- There are a total of 120 counters in a box, consisting only of red and blue counters.
- The red counters are three times the number of blue counters.
2. Set up equations:
- Let's say the number of blue counters is [tex]\( x \)[/tex].
- Then, the number of red counters would be [tex]\( 3x \)[/tex].
- According to the problem, the total number of counters is [tex]\( x + 3x = 4x = 120 \)[/tex].
3. Solve for [tex]\( x \)[/tex]:
- [tex]\( 4x = 120 \)[/tex]
- Divide both sides by 4 to find the number of blue counters:
[tex]\[
x = \frac{120}{4} = 30
\][/tex]
- So, there are 30 blue counters initially.
4. Find the initial number of red counters:
- Since red counters are three times the number of blue counters:
[tex]\[
3x = 3 \times 30 = 90
\][/tex]
- There are 90 red counters initially.
5. Calculate the counters taken:
- Carl takes [tex]\( \frac{1}{3} \)[/tex] of the red counters:
[tex]\[
\frac{1}{3} \times 90 = 30
\][/tex]
So, 30 red counters are taken by Carl.
- Kerry takes [tex]\( 80\% \)[/tex] of the blue counters:
[tex]\[
0.8 \times 30 = 24
\][/tex]
So, 24 blue counters are taken by Kerry.
6. Calculate the counters left:
- Red counters left: [tex]\( 90 - 30 = 60 \)[/tex].
- Blue counters left: [tex]\( 30 - 24 = 6 \)[/tex].
7. Find the ratio of remaining red to blue counters:
- The number of red counters left is 60, and the number of blue counters left is 6.
- The ratio of red counters to blue counters is:
[tex]\[
\frac{60}{6} = 10
\][/tex]
- Therefore, the simplified ratio of the number of red counters to blue counters now in the box is [tex]\( 10:1 \)[/tex].
I hope this explanation helps you understand each step in solving the problem!
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