Question
The track team is trying to reduce their time for a relay race. First they reduce their time by 8 seconds. Then they are able to reduce that time by 25%. If their final time is 51 seconds, what was their beginning time?
Choose the tape diagram that represents the number of seconds in their second time.
x
Time 1
Time 2
x −8
x
Time 1
Time 2
8
x
Excellent!
Choose the tape diagram that represents their final time.
x
Time 1
Time 2
x −8
Time 3
51
x
Time 1
Time 2
x −8
Time 3
51
Good work!
Write an equation in words to represent this situation.
x
Time 1
Time 2
x −8
Time 3
51
125% of Time 2 = Time 3
50% of Time 2 = Time 3
25% of Time 2 = Time 3
75% of Time 2 = Time 3
Choose the tape diagram that represents the number of seconds in their second time.
x
Time 1
Time 2
x −8
x
Time 1
Time 2
8
x
Excellent!
Choose the tape diagram that represents their final time.
x
Time 1
Time 2
x −8
Time 3
51
x
Time 1
Time 2
x −8
Time 3
51
Good work!
Write an equation in words to represent this situation.
x
Time 1
Time 2
x −8
Time 3
51
125% of Time 2 = Time 3
50% of Time 2 = Time 3
25% of Time 2 = Time 3
75% of Time 2 = Time 3
Answers
GPT-4o mini
To determine the original time of the relay race, we can set up the scenario step-by-step.
1. **Starting with the original time, x.**
2. **After the first reduction, the time becomes x - 8 seconds.**
3. **They then reduce this new time (x - 8) by 25%.** To calculate this, we can express the final time (Time 3) after this reduction mathematically.
When the time is reduced by 25%, that means the team keeps 75% of the original reduced time. Therefore, the equation can be expressed as:
\[
\text{Final Time} = \text{Time 2} \times 0.75
\]
or,
\[
51 = (x - 8) \times 0.75
\]
Now, let's express Time 2 (x - 8) in terms of Time 3:
\[
\text{Time 3} = \frac{3}{4} \times \text{Time 2}
\]
Thus, we can write:
\[
51 = 0.75 \times (x - 8)
\]
### Finding the Equation
So, we can respond to your last choice about writing an equation in words:
A correct representation would be:
**75% of Time 2 = Time 3**
Now, to solve for x, we continue:
1. Multiply both sides by 4 to eliminate the fraction:
\[
4 \times 51 = 3(x - 8)
\]
\[
204 = 3(x - 8)
\]
2. Distribute the 3:
\[
204 = 3x - 24
\]
3. Add 24 to both sides:
\[
204 + 24 = 3x
\]
\[
228 = 3x
\]
4. Divide by 3:
\[
x = 76
\]
Thus, the original time (Time 1) was **76 seconds**.
### Summary of Diagrams
- The tape diagram representing the second time is:
- \( x \) as Time 1
- \( x - 8 \) as Time 2
- The tape diagram representing the final time is:
- \( x \) as Time 1
- \( x - 8 \) as Time 2
- \( 51 \) as Time 3
And the equation in words representing this situation is:
**75% of Time 2 = Time 3**.
1. **Starting with the original time, x.**
2. **After the first reduction, the time becomes x - 8 seconds.**
3. **They then reduce this new time (x - 8) by 25%.** To calculate this, we can express the final time (Time 3) after this reduction mathematically.
When the time is reduced by 25%, that means the team keeps 75% of the original reduced time. Therefore, the equation can be expressed as:
\[
\text{Final Time} = \text{Time 2} \times 0.75
\]
or,
\[
51 = (x - 8) \times 0.75
\]
Now, let's express Time 2 (x - 8) in terms of Time 3:
\[
\text{Time 3} = \frac{3}{4} \times \text{Time 2}
\]
Thus, we can write:
\[
51 = 0.75 \times (x - 8)
\]
### Finding the Equation
So, we can respond to your last choice about writing an equation in words:
A correct representation would be:
**75% of Time 2 = Time 3**
Now, to solve for x, we continue:
1. Multiply both sides by 4 to eliminate the fraction:
\[
4 \times 51 = 3(x - 8)
\]
\[
204 = 3(x - 8)
\]
2. Distribute the 3:
\[
204 = 3x - 24
\]
3. Add 24 to both sides:
\[
204 + 24 = 3x
\]
\[
228 = 3x
\]
4. Divide by 3:
\[
x = 76
\]
Thus, the original time (Time 1) was **76 seconds**.
### Summary of Diagrams
- The tape diagram representing the second time is:
- \( x \) as Time 1
- \( x - 8 \) as Time 2
- The tape diagram representing the final time is:
- \( x \) as Time 1
- \( x - 8 \) as Time 2
- \( 51 \) as Time 3
And the equation in words representing this situation is:
**75% of Time 2 = Time 3**.