Chapter Twenty-Three — Word Problems

The Humongous Book of Algebra Problems

534

Express 0.75 as a fraction: .

Multiply the entire equation by 12, the least common denominator, to eliminate

fractions.

Solve for t

2

.

The ﬁrst car will travel hours before the collision.

Mixture and Combination

Measuring ingredients in a mixture

23.30 A 12-ounce cup of fruit punch contains 70% fruit juice and 30% water. If

combined with a 16-ounce bottle of fruit punch that contains 10% fruit juice

and 90% water, what portion of the mixture is juice? Report the answer as a

percentage rounded to the hundredths place.

To solve a mixture problem, multiply the amount of each ingredient by the

value that describes it (such as its concentration) and add the results. Set the

sum equal to the total amount of the mixture multiplied by its descriptive value.

Multiply by 3,600 to convert to

seconds:

.

0.75 is

read

“seventy-ve

HUNDREDths.”

To convert the

decimal into a

fraction, divide

75 by one

HUNDRED.

Chapter Twenty-Three — Word Problems

The Humongous Book of Algebra Problems

535

In this problem, the ingredients are 12- and 16-ounce containers of punch,

which contain 70% and 10% real juice, respectively. Convert the percentages

into decimals (70% = 0.7 and 10% = 0.1) and substitute the values into the

equation blueprint above. Let x represent the unknown value, the portion of the

ﬁnal mixture that is real juice.

The mixture is approximately 35.71% fruit juice.

23.31 If a 5-pound bag of mixed nuts that is 45% peanuts by weight is combined with

a 12-pound bag of mixed nuts that is 40% peanuts by weight, what percentage

of the mixture is peanuts (by weight)? Report the answer as a percentage

rounded to the hundredths place.

Apply the blueprint for mixture and combination problems.

Multiply the ﬁrst ingredient, a 5-pound bag, by 0.45 (its descriptive value 45%

expressed as a decimal). Multiply the second ingredient, a 12-pound bag, by its

descriptive value, 0.4. The mixture weighs 5 + 12 = 17 pounds; let x represent

the unknown descriptive value, the percentage of the mixture that is peanuts by

weight.

The mixture is approximately 41.47% peanuts by weight.

23.32 How much extra-virgin olive oil, containing 0.8% acidity, must be added to 4

gallons of olive oil with an acidity of 2.5% so that the mixture has an acidity of

2%? Report the answer in gallons, rounded to the hundredths place.

Apply the blueprint for mixture and combination problems.

To convert

from a decimal

to a percent, move

the decimal point two

places to the right.

Don’t round to the

hundredths place until

after you move the

decimal.

Chapter Twenty-Three — Word Problems

The Humongous Book of Algebra Problems

536

Let x represent the unknown amount (in gallons) of the ﬁrst ingredient, the

extra-virgin olive oil; multiply x by its acidity (0.008). Add the amount of the

second ingredient (4 gallons) multiplied by its acidity (0.025). The total mixture

is x + 4 gallons and has an acidity of 0.02.

Solve for x.

In order to give the mixture an acidity of 2%, 1.67 gallons of extra-virgin olive

oil are required.

23.33 Two machines at a manufacturing plant are used to fulﬁll an order for

widgets. Machine A produces 350 widgets, and machine B produces 800.

Upon receiving the order, the customer notes that 138 of the widgets have

manufacturing defects.

If engineers determine that approximately 1.5% of the widgets made by

machine B are defective, what percentage of the widgets made by machine A

are defective?

Let w

A

= 350 represent the number of widgets made by machine A, w

B

= 800

represent the number of widgets made by machine B, r

A

represent the

percentage of widgets made by machine A that contain defects, r

B

= .015

represent the percentage of widgets made by machine B that contain defects,

w

o

= 800 + 350 = 1,150 represent the total number of widgets ordered, and

r

o

= 138 ÷ 1,150 = 0.12 represent the percentage of the widgets ordered that were

defective. Apply the blueprint for mixture and combination problems.

Solve for r

A

.

Approximately 36% of the widgets made by machine A were defective.

Even though

the percentage

0.8% contains a

decimal, you still

need to move that

decimal point two

places to the left

to convert the

percentage into a

decimal value:

0.8% = 0.008.

138 out of the

1,150 total widgets

had defects, which is

0.12 = 12% of the order.

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