Word Problems

Word problems test translation before they test math: rates and work, mixtures, overlapping sets, age and interest problems, and weighted averages. The arithmetic inside is rarely hard — the challenge is converting three sentences of story into equations quickly and without inventing constraints the problem never stated. Practice writing the setup in a fixed, boring format every time (define variables, one equation per fact), and learn the standard templates: rate × time = work, the two-set matrix, and the teeter-totter method for weighted averages.

Mixtures, rates, distances, ages.·Show:20 30 50

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505-555 (Easy)

Shawana made a schedule for reading books during 4 weeks (28 days) of her summer vacation. She has checked out 12 books from the library. The number of pages in each book and the order in which she plans to read the books are shown in the table above. She will read exactly 50 pages each day. The only exception will be that she will never begin the next book on the same day that she finishes the previous one, and therefore on some days she may read fewer than 50 pages. At the end of the 28th day, how many books will Shawana have finished?

505-555 (Easy)

What number is

108

more than two-thirds of itself?

505-555 (Easy)

A service provider charges

c

dollars for the first 50 hours of service used per month and 40 cents for each 30 minutes in excess of 50 hours used during the month. If

x

is an integer greater than 50, which of the following expressions gives this service provider’s charge, in dollars, for a month in which

x

hours of service were used?

505-555 (Easy)

A dance troupe has a total of 50 dancers split into 2 groups. The costumes worn by Group A cost

$80

each, and those worn by Group B cost

$90

each. If the total cost of all the costumes is

$4, 270

, what is the total cost of the costumes worn by Group B?

505-555 (Easy)

An auction house charges a commission of 15 percent on the first

$50, 000

of the sale price of an item, plus 10 percent on the amount of the sale price in excess of

$50, 000

. What was the sale price of a painting for which the auction house charged a total commission of

$24, 000

505-555 (Easy)

A certain method for calculating pediatric drug dosages states that if

d

is the adult dosage, in milligrams, and

y

is the child’s age, in years, then the child’s dosage, in milligrams, is

d (\frac{y + 1}{24})

. If the adult dosage for a particular drug is 800 milligrams, what is an 8-year-old child’s dosage, in milligrams, according to this method?

605-655 (Medium)

To rent a tractor, it costs a total of

x

dollars for the first 24 hours, plus

y

dollars per hour for each hour in excess of 24 hours. Which of the following represents the cost, in dollars, to rent a tractor for 36 hours?

605-655 (Medium)

The profit

P

, in dollars, for any given month at a certain company is defined by

P = I - C

, where

I

represents total income and

C

represents total costs for the month. For each of the first 4 months,

C = I + 32, 000

; for each of the next 3 months,

I = C + 36, 000

; and for each of the 5 remaining months,

I = C + 10, 000

. What was the company’s total profit for the 12-month year?

605-655 (Medium)

A worker carries 4 jugs per trip. If the jugs are packed into cartons that hold 7 jugs each, how many jugs are needed to fill the last partially filled carton after the worker has made 17 trips?

605-655 (Medium)

Last year a state senate consisting only of Republican and Democrat members had 20 more Republican members than Democrat members. This year it has the same total membership but 2 fewer Republican members. If this year the number of Republican members is

\frac{2}{3}

the number of senate members, how many members does the senate have this year?

605-655 (Medium)

Sam has

$800

in his account. He deposits

$1

one week from now,

$2

two weeks from now, and each week thereafter

$1

more than the previous week. With no other transactions, how much will he have 50 weeks from now?

605-655 (Medium)

A certain state’s milk production was 980 million pounds in 2007 and 2.7 billion pounds in 2014. Approximately how many more million gallons did it produce in 2014? (

1

billion

= 1 0^{9}

and 1 gallon = 8.6 pounds.)

605-655 (Medium)

The cost

C

, in dollars, to remove

p

percent of a pollutant is

C = \frac{100 , 000 p}{100 - p}

. How much more would it cost to remove 90 percent than 80 percent?

605-655 (Medium)

Mark and Ann together were allocated

n

boxes of cookies to sell for a club project. Mark sold 10 boxes less than

n

and Ann sold 2 boxes less than

n

. If Mark and Ann have each sold at least one box of cookies, but together they have sold less than

n

boxes, what is the value of

n

605-655 (Medium)

Exchange Rates in a Particular Year

$1 = 5.3

francs

$1 = 1.6

marks An American dealer bought a table in Germany for 480 marks and sold the same table in France for 2,385 francs. What was the dealer’s gross profit on the two transactions in dollars?

605-655 (Medium)

Last year Brand X shoes were sold by dealers in 403 different regions worldwide, with an average (arithmetic mean) of 98 dealers per region. If last year these dealers sold an average of 2,488 pairs of Brand X shoes per dealer, which of the following is closest to the total number of pairs of Brand X shoes sold last year by the dealers worldwide?

655-705 (Hard)

The total cost for Company X to produce a batch of tools is

$10, 000

plus

$3

per tool. Each tool sells for

$8

. The gross profit earned from producing and selling these tools is the total income from sales minus the total production cost. If a batch of 20,000 tools is produced and sold, then Company X’s gross profit per tool is

655-705 (Hard)

If Jake loses 8 pounds, he will weigh twice as much as his sister. Together they now weigh 278 pounds. What is Jake’s present weight, in pounds?

655-705 (Hard)

The light in a restroom operates with a 15-minute timer that is reset every time the door opens as a person goes in or out of the room. Thus, after someone enters or exits the room, the light remains on for only 15 minutes unless the door opens again and resets the timer for another 15 minutes. If the times listed above are the times at which the door opened from 8:00 to 10:00, approximately how many minutes during this two-hour period was the light off?

655-705 (Hard)

To mail a package, the rate is

x

cents for the first pound and

y

cents for each additional pound, where

x > y

. Two packages weighing 3 pounds and 5 pounds, respectively, can be mailed separately or combined as one package. Which method is cheaper, and how much money is saved?

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