HSPT Math - How to do work problems

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.

Explanation

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.

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?

Explanation

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 .

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?

The side length and of the garden and the shape of the garden.

Ten tulips

Three roses

Four petunias

Three lilies

Explanation

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.

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?

$4500

$3500

$7000

$8000

$1500

Explanation

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.

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?

280

200

120

960

Explanation

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:

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

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.

Explanation

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.

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?

Explanation

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

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?

$83 \frac{1}{3}$

$33\frac{1}{3}$

Explanation

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

loaves of bread.

Divide 10,000 loaves by 500 loaves per hour:

$10,000 \div 500 = 20$ .

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?

Two hours and forty-six minutes

Three hours

Three hours and twenty-six minutes

Two hours

Three hours and six minutes

Explanation

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.

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.