Math / Algebra

QuestionDuring a record cold-snap, Riverton High School's boilers give out. Ordinarily the repairs would take Mr Thompson 8 hours. To complete the project faster, Principal Waite deputizes Vice-Principals Tranter, Richins, and Tilby as emergency mechanics. (They may lack training and talent, but their enthusiasm is beyond question!) If Mr Thompson's new assistants each work at half his speed, how many hours will it take to repair the boilers?
Total Time $=$ $\square$ Answers may be written as unsimplified fractions or as decimals. If you choose decimals, round your answer

Studdy Solution

STEP 1

What is this asking? How long will it take to fix the boilers if Mr.
Thompson gets three helpers who are half as fast as him? Watch out! Don't just add the times together!
More people means less time.

STEP 2

1. Calculate the helpers' speed.
2. Calculate the combined speed.
3. Calculate the total time.

STEP 3

Alright, so Mr.
Thompson can fix the boilers in 8 hours by himself.
That means his speed is $\frac{1 \text{ job}}{8 \text{ hours}}$ , or $\frac{1}{8}$ of the job per hour.
We're dividing to one here to get the speed in terms of jobs per hour!

STEP 4

Each helper works at half his speed.
So, each helper's speed is $\frac{1}{2} \cdot \frac{1}{8} = \frac{1}{16}$ of the job per hour.
We multiply by one-half to get half the speed.

STEP 5

There are three helpers, so their combined speed is $3 \cdot \frac{1}{16} = \frac{3}{16}$ of the job per hour.
We multiply by three because there are three helpers.

STEP 6

Now, let's add Mr.
Thompson's speed to the helpers' combined speed!
Mr. Thompson's speed is $\frac{1}{8}$ , which is the same as $\frac{2}{16}$ .
So, the total combined speed of everyone working together is $\frac{3}{16} + \frac{2}{16} = \frac{3+2}{16} = \frac{5}{16}$ of the job per hour.
We're adding to get the total speed!

STEP 7

If they complete $\frac{5}{16}$ of the job every hour, how long will it take to complete the whole job (which is equivalent to 1 job)?
To find out, we divide the total job (1) by the combined speed ( $\frac{5}{16}$ ).
Think of it like this: if you travel at 50 miles per hour, and you need to go 100 miles, it will take you $100 \div 50 = 2$ hours.

STEP 8

So, the total time is $\frac{1}{\frac{5}{16}}$ .
To divide by a fraction, we multiply by its reciprocal (flip it!), which gives us $1 \cdot \frac{16}{5} = \frac{16}{5}$ hours.
We multiply by the reciprocal to divide by a fraction.

STEP 9

We can also write this as a mixed number: $\frac{16}{5} = 3\frac{1}{5}$ hours, or as a decimal: $\frac{16}{5} = 3.2$ hours.

STEP 10

It will take $\frac{16}{5}$ hours, or 3.2 hours, to fix the boilers.

Was this helpful?

Studdy Solution

STEP 1

What is this asking? How long will it take to fix the boilers if Mr.
Thompson gets three helpers who are half as fast as him? Watch out! Don't just add the times together!
More people means less time.

STEP 2

1. Calculate the helpers' speed.
2. Calculate the combined speed.
3. Calculate the total time.

STEP 3

Alright, so Mr.
Thompson can fix the boilers in 8 hours by himself.
That means his speed is $\frac{1 \text{ job}}{8 \text{ hours}}$ , or $\frac{1}{8}$ of the job per hour.
We're dividing to one here to get the speed in terms of jobs per hour!

STEP 4

Each helper works at half his speed.
So, each helper's speed is $\frac{1}{2} \cdot \frac{1}{8} = \frac{1}{16}$ of the job per hour.
We multiply by one-half to get half the speed.

STEP 5

There are three helpers, so their combined speed is $3 \cdot \frac{1}{16} = \frac{3}{16}$ of the job per hour.
We multiply by three because there are three helpers.

STEP 6

STEP 7

STEP 8

So, the total time is $\frac{1}{\frac{5}{16}}$ .
To divide by a fraction, we multiply by its reciprocal (flip it!), which gives us $1 \cdot \frac{16}{5} = \frac{16}{5}$ hours.
We multiply by the reciprocal to divide by a fraction.

STEP 9

We can also write this as a mixed number: $\frac{16}{5} = 3\frac{1}{5}$ hours, or as a decimal: $\frac{16}{5} = 3.2$ hours.

STEP 10

It will take $\frac{16}{5}$ hours, or 3.2 hours, to fix the boilers.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

What is this asking? How long will it take to fix the boilers if Mr.Thompson gets three helpers who are half as fast as him? Watch out! Don't just add the times together!More people means *less* time.

STEP 2

1. Calculate the helpers' speed.2. Calculate the combined speed.3. Calculate the total time.

STEP 3

Alright, so Mr.Thompson can fix the boilers in **8 hours** by himself.That means his speed is 1 job8 hours\frac{1 \text{ job}}{8 \text{ hours}}8 hours1 job​, or 18\frac{1}{8}81​ of the job per hour.We're dividing to one here to get the speed in terms of jobs per hour!

STEP 4

Each helper works at *half* his speed.So, each helper's speed is 12⋅18=116\frac{1}{2} \cdot \frac{1}{8} = \frac{1}{16}21​⋅81​=161​ of the job per hour.We multiply by one-half to get half the speed.

STEP 5

There are *three* helpers, so their *combined* speed is 3⋅116=3163 \cdot \frac{1}{16} = \frac{3}{16}3⋅161​=163​ of the job per hour.We multiply by three because there are three helpers.

STEP 6

STEP 7

STEP 8

So, the total time is 1516\frac{1}{\frac{5}{16}}165​1​.To divide by a fraction, we multiply by its reciprocal (flip it!), which gives us 1⋅165=1651 \cdot \frac{16}{5} = \frac{16}{5}1⋅516​=516​ hours.We multiply by the reciprocal to divide by a fraction.

STEP 9

We can also write this as a mixed number: 165=315\frac{16}{5} = 3\frac{1}{5}516​=351​ hours, or as a decimal: 165=3.2\frac{16}{5} = 3.2516​=3.2 hours.

STEP 10

It will take 165\frac{16}{5}516​ hours, or 3.2 hours, to fix the boilers.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? How long will it take to fix the boilers if Mr.
Thompson gets three helpers who are half as fast as him? Watch out! Don't just add the times together!
More people means less time.

1. Calculate the helpers' speed.
2. Calculate the combined speed.
3. Calculate the total time.

Alright, so Mr.
Thompson can fix the boilers in 8 hours by himself.
That means his speed is $\frac{1 \text{ job}}{8 \text{ hours}}$ , or $\frac{1}{8}$ of the job per hour.
We're dividing to one here to get the speed in terms of jobs per hour!

Each helper works at half his speed.
So, each helper's speed is $\frac{1}{2} \cdot \frac{1}{8} = \frac{1}{16}$ of the job per hour.
We multiply by one-half to get half the speed.

There are three helpers, so their combined speed is $3 \cdot \frac{1}{16} = \frac{3}{16}$ of the job per hour.
We multiply by three because there are three helpers.

So, the total time is $\frac{1}{\frac{5}{16}}$ .
To divide by a fraction, we multiply by its reciprocal (flip it!), which gives us $1 \cdot \frac{16}{5} = \frac{16}{5}$ hours.
We multiply by the reciprocal to divide by a fraction.

We can also write this as a mixed number: $\frac{16}{5} = 3\frac{1}{5}$ hours, or as a decimal: $\frac{16}{5} = 3.2$ hours.

It will take $\frac{16}{5}$ hours, or 3.2 hours, to fix the boilers.