CALCULATOR WORD PROBLEMS

Objective:  To learn how to not fear complicated calculations in word problems that may require a calculator.

Remark The lessons that you learned in the last hour and last day are here reinforced through exercises that combine calculator activity and word problem solving.  There is really nothing new to recall in this hour.  The recall and exercise session is small, allowing more time for you to work on the problems themselves.

Example 1.    The length of an equilateral triangle is equal to four times the length of one of its sides.  If the perimeter of the triangle is 490 meters, what is the length of one of its sides?

Solution:    Make a sketch of the rectangle and label a side.

We know that the perimeter of the rectangle (the distance around the rectangle) is 490 meters.  Let P = the perimeter.  Then

P = 4x = 490
So,    x = 490,
and    x = 490/4

Therefore, one side is 122.5 meters.

Example 22.2. (Percent Problem)
In an experiment, it is calculated that in advertising a new soap on TV, 67.25% of the people who bought the soap, had done so because they had seen the advertising campaign more than on one occasion.  If 1,756,000 people had seen the ad on more than one occasion, how many people bought the soap?

Solution: [Remember that percent means hundredths.  So, if you say 67.25 percent you mean 67.25 hundredths or 67.25/100.   This means that if you say

67.25 percent of x    you mean  67.25/100  times x ].

Let x be the number of people who bought the soap.

We know that (67.25/100) x = 1,756,000.

Solving for x we find that x = 2.611152 X 10^6 people bought the soap.

Note 1
Scientific notation makes it possible for you to use a computer or pocket calculator to work with very large or very small numbers.  Most calculators do not display more than 9 digits and most computers will not evaluate numbers having more than 12 digits.  The number 3.4598 / 10^15 is so small, the computer will consider it to be--for all practical purposes--zero.  Scientific notation fools the calculator into thinking that the number being entered is simply a number between 1 and 10.

Example 3. (percentage)

A three year drought causes some crop failure in a midwestern corn field.  One acre of land can yield 2540 bushels of corn.  In a typical year 23.55% of the crops are damaged.  Another 13.23% are damaged on route to the market, in an average year.  If the farmer sells 220,455 bushels, how many acres of corn should be planted?

Solution: Let x = the number of bushels that should be planted.

23.55% of x is the number of crops that did not survive.  Hence,
(23.55/100)x = the number of crops that did not survive.

13.23% of x is the number of crops damaged on the way to   market.

(13.23/100)x = the number of crops damaged on the way to   market.

Use the fact that

number of bushels to be planted
minus
number of bushels that did not survive
minus
number of bushels that were damaged
equals
220,455

This translate into: x - (23.55/100)x - (13.23/100)x = 220,455

Solving for x (by first multiplying by 10000),we get

10000x - 2355x - 1323x = 2,204,550,000,
or  6322x = 2,204,550,000,

x =348,711

This number represents the number of bushels of corn that should be planted.  The exercise calls for the number of acres needed to plant the corn.   Remember that each acre will yield 2,540 bushels of corn, so one bushel will need 1/2540 acres.

If 1 bushel = 1/2540 acres, then

348,711 bushels = 348,711*(1/2540) acres.

= 1.3728779 x10^2 acres.

Example 4.  (Mixture problem)
An octane rating of a gasoline is the percentage by volume of isooctane, an antiknock ingredient usually mixed into the gasoline.  An oil company uses its refinery to mix 23,400 gallons of 87.5 octane gas with 18,300 gallons of 92.2 octane gas.  What is the octane of the mixture?

Solution: Let x be the octane of the mixture.

We use the fact that
octane of one gas      +     octane of second gas     =    octane of mixture
87.5% of 23,400       +        94.2% of 18,300       =  x% of (23,400+18,300/)

The quantity 23,400 + 18,300   comes from the fact that    the new mixture will    contain 23,400 + 18,300    gallons of gasoline.

(87.5/100)*23,400   +    ( 94.2/100) *18,300         =  (x/100)*41,700

Or,   (87.5)(234) + (92.2)(183) = 417x,
20475 + 16872.6 = 417x,
37347.6 = 417x,
x =  approximately   89.56 octane.

Example 5.   Two grasshoppers race through a meadow 455 feet wide.  The winner came in only 12 seconds after the looser.  If the winner averaged 3.45 hops per second during the race, how much faster (on average) should the looser have gone in order to win the race?

Solution: Let x = the amount faster that the looser needed to go in order to win the race.

We use the relation between distance, velocity and time as follows:

distance = velocity x time.

The total distance of the race is 455 feet.  Therefore,

3.45 x t = 455 feet

where t is winner's time in finishing the race (in seconds).

We know that the looser came in 12 seconds after the winner.  Hence, we know that the velocity of the looser times (t + 12) = 455.  In other words, if we let vL = 3.45 hops per second = the velocity of the looser, then

vL*(t + 12) = 455.

We may find t by solving the equation

3.45(t + 12) = 455

for t.  We find that t = 119.88405 seconds.  Therefore, if we

(vL + x)(119.88405) = 455.

We know that vL = 3.45 so the equation is

(3.45 + x)  (119.88405) = 455,
413.59997 + 119.88405*x = 455,
119.88405*x = 41.4.

Solving for x, we get x ª 0.345 hops per second.

EXERCISES

1. Two numbers have a relationship where one minus 3.4 times the other equals 4.5  106..  The average of the two numbers is 3.3  105.  Find the numbers.

ans  -2.81x 10^6..and -2.15 x10^6.

2. An aging king decided that it was time to divide his vast land among his three daughters.   He gave 28% to one daughter, 32% to another and 35% to the third.  That left him with a meager 2,324 acres on which he could still hunt fox.  How much land did he leave to each daughter?

ans  13,014.4 acres, 14,873.6 acres, and 16,268 acres

3  A creamery inspects all the milk it processes, rejecting on a daily average 1.54% of the total amount that it processes.  How many gallons does it reject in a year, if it accepts 985 gallons a day?

ans 5,623.28 gallons per year

4 Two reservoirs are joined by a pipeline that mixes the water and brings the mixture to a third reservoir.  One contains 155,250 gallons with salt at a concentration 0.45%.  The other contains 267,500 gallons of pure water (no salt).  Will there be enough pure water to make 422,750 gallons of 0.155% salt water?

ans  No    Need another 27,975.8 gallons of pure water.

5 Two fighter planes are a distance D apart and are approaching each other.  .  One is traveling at the rate of 1467 feet per second, while the other is traveling at 1173 feet per second.  If it takes two minutes for them to be within 1000 feet from each other, what is D?

ans  D = 317,800 feet  or about 60 miles.

6 An automobile advertisement gives the prices of three different engine options for the same car.  One car has a four cylinder engine with an overhead cam and costs \$7,449.00.  Another has a four cylinder engine with an overhead cam and costs \$11,089.00.  A third has a V8 engine with no overhead cam.  It costs \$12,129.00.  If the costs are entirely due to the engines and cams, what does a cam cost?  What does a V8 engine cost?

ans  .

7 Two fighter planes depart from an aircraft carrier and head in opposite directions.  One plane has a velocity of 34 miles per hour faster than the other  After 8 minutes they are 140 miles apart.  Find the speed of each plane.

ans   508 mph and 542 mph

8 A navy F111 fighter plane is 1500 yards from the landing strip of an aircraft carrier and on its way in for a landing.  If the ship is traveling north at 24 miles per hour and the plane is traveling north at 234 miles per hour, how long will it be before the plane can land?

ans   approximately 14.6 seconds.