How to do work problems - HSPT Math

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Question

Steve can paint his greenhouse in four hours working alone; his ten-year-old son can paint the greenhouse alone in nine hours. Working together, how long, to the nearest minute, would it take them?

Answer

Think of these work rates as "greenhouses per hour", not "hours per greenhouse". Then Steve's work rate is one fourth of a greenhouse per hour, and his son's is one ninth of a greenhouse per hour. Now, multiply each rate by the time to get the portion of the greenhouse each paints, and add the products up to get one greenhouse. The work equation becomes:

$\frac{1}{4} t + \frac{1}{9} t = 1$

$\frac{9}{36} t + \frac{4}{36} t = 1$

$\frac{13}{36} t = 1$

$\frac{36}{13}\cdot \frac{13}{36} t = \frac{36}{13}\cdot 1$

$t = \frac{36}{13} ;\textrm{hrs}$

or $\frac{36}{13} \cdot 60 \approx 166; \textrm{min}$

This is two hours and forty-six minutes.

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Question

Bob gets paid an hour for the regular hours he works and an hour for any overtime hours he works. All hours over 40 in a week are considered overtime. If Bob works 44 hours this week, how much did he make?

Answer

You first calculate how much he makes for normal hours, which is the number of hours works multiplied by the wage. So for normal hours, $40\times5.25 = $210$ . For overtime, it is $4\times6 = 24$ . Add the amount made in overtime and the normal hours and you get .

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Question

Julie sells cars. She has a monthly salary of $1000 and a commission of $500 for each car sold. How much money will she make this month if she sells 7 cars?

Answer

To find out how much Julie makes from selling the cars, we first multiply the 7 cars she sells by the $500 that she receives for each car. This gives us $3,500. We now add the $1,000 that she gets for her monthly salary for a total of $4,500.

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Question

Machine A can produce 4 books in 2 days. Machine B can produce 20 books in 4 days. How many books can machines A and B, working together, produce in 40 days?

Answer

Through simplification, we can see that machine A produces 2 books a day and that machine B produces 5 books a day. If we multily each by 40 days and add them together, we find that together they can produce 280 books in 40 days:

$(2+5)\times 40=7\times 40=280$

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Question

John sells cloths. His monthly salary is dollars plus a commission of dollars for each cloth sold. If he sells cloths this month, how much money does he make in dollars?

Answer

In order to find how much he makes from selling the cloths, we first multiply the number of cloths sold by dollars.

$80\times 7=560$ dollars

Now we add his monthly salary:

dollars

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Question

Tom works in a mall and sells shoes. His monthly salary is dollars plus a commission of dollars for every pair of shoes that he sells. How many pairs of shoes must he sell to earn dollars per month.

Answer

Let number of pairs of shoes.

Then we can set up the following equation:

Now we solve the equation for :

$\Rightarrow 2060-1100=(1100-1100)+8x\Rightarrow 960=8x\Rightarrow x=\frac{960}{8}=120$

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Question

John and Tom are farmers. John can produce $1500\ lb$ of an organic product in six months. Tom can produce $3500\ lb$ of the organic product in one year. How many pounds of organic product can they produce in three years together? (Assume that they can work and produce the organic product all year round.)

Answer

John can produce $1500\ lb$ in six month. Therefore we can set up a proportion:

$\frac{1500 \ lb}{0.5\ yr}=\frac{x \ lb}{3\ yr}\Rightarrow 0.5x=3\times 1500\Rightarrow x=9000\ lb$

Tom can produce $3500\ lb$ in one year. Then we can set up another proportion:

$\frac{3500 \ lb}{1\ yr}=\frac{y \ lb}{3\ yr}\Rightarrow 1\times y=3\times 3500 \Rightarrow y=10500\ lb$

$x+y=9000+10500=19500\ lb$

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Question

A bakery has a machine that can make two hundred loaves of bread in one hour, and a machine that can make three hundred loaves of bread in one hour. How many hours will it take for the two machines together to make 10,000 loaves of bread?

Answer

In one hour, the two machines can make a total of

loaves of bread.

Divide 10,000 loaves by 500 loaves per hour:

$10,000 \div 500 = 20$ .

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Question

A candymaker has two machines. One can turn out one hundred pieces of divinity in one hour; the other, an older machine, can do the same in one hour and twenty minutes. Working together and starting at the same time, how many hours would it take for them to make 10,000 pieces of divinity?

Choose the closest answer.

Answer

The newer machine can make 100 pieces of divinity per hour. The older machine makes 100 pieces in one hour twenty minutes, or, equivalently, one and one third hours, which is

$100 \div 1 \frac{1}{3} = 100 \div \frac{4}{3} = 100 \times \frac{3}{4} = 75$ pieces per hour.

Together, the machines make pieces per hour.

Divide 10,000 pieces by 175 pieces per hour:

$10,000 \div 175 = 57 \textup{ R }1$ .

The candy can be made in a little over 57 hours. The closest choice is therefore 60 hours.

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Question

A candymaker specializes in making "turtles" (pecan caramel candies). She has two machines - a newer machine, which can make one hundred turtles in seventy minutes, and an older machine, which can make one hundred turtles in ninety minutes.

She wants to make one hundred boxes of turtles, each of which holds forty turtles. How many hours would it take for both machines, working together, to produce enough turtles to fill those boxes?

Choose the closest answer.

Answer

The newer machine makes one hundred turtles in seventy minutes, or, on the average,

$\frac{100}{70} = \frac{10}{7}$ turtles in one minute.

Similarly, the older machine makes, on the average,

$\frac{100}{90} = \frac{10}{9}$ turtles in one minute.

Together, they make an average of

$\frac{10}{7} + \frac{10}{9} = \frac{10 \times 9 }{7\times 9 } + \frac{7\times 10}{7\times 9} = \frac{ 90 }{63 } + \frac{7 0}{63} = \frac{160 }{63 }$

turtles per minute.

The candymaker wants one hundred boxes, each with forty turtles, so she wants

$100 \times 40 = 4,000$ turtles. Divide this by $\frac{160 }{63 }$ turtles per minute to get

$4,000 \div \frac{160 }{63 } = 4,000 \times \frac{63 } {160 }$ minutes;

multiply this by $\frac{1}{6}$ to get the number of hours:

$4,000 \times \frac{63 } {160 } \times \frac{1}{60 }= \frac{252,000}{9,600} = 26 \textup{ R }2400$

The process takes just over 26 hours; the response closest to the correct time is 25 hours.

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Question

A machine can make one hundred caramel creams in twenty five minutes.

The owner of the machine wants to fill an order of two hundred boxes of caramel creams; eighty of the boxes are to be large boxes, which contain fifty candies each, and the rest are to be small boxes, which contain thirty candies each. How many hours will it take to make enough candies to fill all two hundred boxes?

Select the closest answer.

Answer

80 boxes contain 50 candies each; 120 boxes contain 30 candies each. Therefore, there will be a total of

$80 \times 50 + 120 \times 30 = 4,000 + 3,600 = 7,600$

candies in the boxes.

It takes 25 minutes to produce 100 candies. It takes

$25 \times \frac{7,600}{100 } = 25 \times 76 = 1,900$

minutes to produce 7,600 candies. Divide by 60 to convert to hours:

$1,900 \div 60 = 31 \textup{ R }40$

This is 31 hours and 40 minutes. The closest response to this is 30 hours.

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Question

Phillip can paint his father's greenhouse in six hours, working alone. His brother Greg can perform the same task in hours. If the two work together, how many hours, in terms of , does it take them to complete the task?

Answer

Phillip can paint the greenhouse in six hours, so he can paint $\frac{1}{6}$ of the greenhouse in one hour. Likewise, Greg can paint the greenhouse in hours, so he can paint $\frac{1}{T}$ of the greenhouse in one hour.

Working together, they can paint

$\frac{1}{6} + \frac{1}{T} = \frac{T}{6T} + \frac{6}{6T} = \frac{T + 6}{6T}$

of the greenhouse in one hour. Take the reciprocal of this; it takes them

$\frac{T + 6}{6T}$

hours to paint the enitre greenhouse, working together.

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Question

Kelly can paint her father's shed in five hours, working alone. Working together, she and her sister Sharon can complete the task in hours. In terms of , how many hours would it take for Sharon to complete this task?

Answer

Let be the number of hours it would take Sharon to paint the shed alone. Kelly alone can paint the shed in hours, so she can paint $\frac{1}{T}$ of a shed per hour. Similarly, Sharon alone can paint $\frac{1}{S}$ shed per hour, so, working together, they can paint

$\frac{1}{S} + \frac{1}{T}$

shed per hour.

Since it takes them five hours to paint the shed together, they can paint $\frac{1}{5}$ shed per hour, so

$\frac{1}{S} + \frac{1}{T} = \frac{1}{5}$

$\frac{1}{S} + \frac{1}{T} - \frac{1}{T} = \frac{1}{5}- \frac{1}{T}$

$\frac{1}{S} = \frac{1}{5}- \frac{1}{T} = \frac{T}{5T}- \frac{5}{5T} = \frac{T- 5}{5T}$

Take the reciprocal of both expressions:

$S = \frac{5T} {T- 5}$ , the correct choice.

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Question

What information in the following word problem is not relevant to the questions?

Jess has a square garden with a length of 5 feet in which there are four kinds of flowers. There are ten tulips, three roses, four petunias, and three lilies. What is the probability of picking a tulip?

Answer

For this problem, first identify what the question is asking for.

"What is the probability of picking a tulip?"

Next, identify what is given.

Shape of the garden: Square

Length of the garden: 5 feet

# of tulips: 10

# of roses: 3

# of petunias: 4

# of lilies: 3

Now, identify what is needed to solve the question.

Since you are looking for the probability of picking a flower, you only need the number of the different kinds of flowers and the total number of flowers to get that answer.

$\textup{Probability}=\frac{#\textup{Tulips}}{\textup{Total of all flowers}}$

From here, find which information that is given is not being used to solve the question.

The side length and of the garden and the shape of the garden would not help find that probability.

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Question

Steve can paint his greenhouse in four hours working alone; his ten-year-old son can paint the greenhouse alone in nine hours. Working together, how long, to the nearest minute, would it take them?

Answer

Think of these work rates as "greenhouses per hour", not "hours per greenhouse". Then Steve's work rate is one fourth of a greenhouse per hour, and his son's is one ninth of a greenhouse per hour. Now, multiply each rate by the time to get the portion of the greenhouse each paints, and add the products up to get one greenhouse. The work equation becomes:

$\frac{1}{4} t + \frac{1}{9} t = 1$

$\frac{9}{36} t + \frac{4}{36} t = 1$

$\frac{13}{36} t = 1$

$\frac{36}{13}\cdot \frac{13}{36} t = \frac{36}{13}\cdot 1$

$t = \frac{36}{13} ;\textrm{hrs}$

or $\frac{36}{13} \cdot 60 \approx 166; \textrm{min}$

This is two hours and forty-six minutes.

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Question

Bob gets paid an hour for the regular hours he works and an hour for any overtime hours he works. All hours over 40 in a week are considered overtime. If Bob works 44 hours this week, how much did he make?

Answer

You first calculate how much he makes for normal hours, which is the number of hours works multiplied by the wage. So for normal hours, $40\times5.25 = $210$ . For overtime, it is $4\times6 = 24$ . Add the amount made in overtime and the normal hours and you get .

Compare your answer with the correct one above

Question

Julie sells cars. She has a monthly salary of $1000 and a commission of $500 for each car sold. How much money will she make this month if she sells 7 cars?

Answer

To find out how much Julie makes from selling the cars, we first multiply the 7 cars she sells by the $500 that she receives for each car. This gives us $3,500. We now add the $1,000 that she gets for her monthly salary for a total of $4,500.

Compare your answer with the correct one above

Question

Machine A can produce 4 books in 2 days. Machine B can produce 20 books in 4 days. How many books can machines A and B, working together, produce in 40 days?

Answer

Through simplification, we can see that machine A produces 2 books a day and that machine B produces 5 books a day. If we multily each by 40 days and add them together, we find that together they can produce 280 books in 40 days:

$(2+5)\times 40=7\times 40=280$

Compare your answer with the correct one above

Question

John sells cloths. His monthly salary is dollars plus a commission of dollars for each cloth sold. If he sells cloths this month, how much money does he make in dollars?

Answer

In order to find how much he makes from selling the cloths, we first multiply the number of cloths sold by dollars.

$80\times 7=560$ dollars

Now we add his monthly salary:

dollars

Compare your answer with the correct one above

Question

Tom works in a mall and sells shoes. His monthly salary is dollars plus a commission of dollars for every pair of shoes that he sells. How many pairs of shoes must he sell to earn dollars per month.

Answer

Let number of pairs of shoes.

Then we can set up the following equation:

Now we solve the equation for :

$\Rightarrow 2060-1100=(1100-1100)+8x\Rightarrow 960=8x\Rightarrow x=\frac{960}{8}=120$

Compare your answer with the correct one above

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