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.
Chapter Twenty-Three — Word Problems
The Humongous Book of Algebra Problems
537
23.34 A collection of 49 nickels and pennies is worth 85 cents. How many of the
coins are nickels?
Let n represent the number of nickels. The collection contains 49 total coins,
so there are p = 49 – n pennies. Multiply the total number of each coin by its
worth (expressed as a decimal) and set the sum equal to 0.85 = 85 cents.
Solve for n.
There are nine nickels among the 49 coins.
23.35 A jar of dimes and quarters is worth $32.45. If the number of quarters is 13 less
than three times the number of dimes, how many quarters are in the jar?
Let d represent the number of dimes in the jar and q = 3d – 13 represent the
number of quarters. Multiply the total number of each coin by its value and set
the sum equal to $32.45.
Solve for d.
The collection of coins contains 42 dimes. Calculate the number of quarters.
The collection of coins contains 113 quarters.
If there are
n nickels and
a total of 49
coins overall, then
subtract the number
of nickels from 49 to
nd out how many
pennies there
are.
3d – 13 is
thirteen less than
three times the
number of dimes.
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