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Word problem (задачи)

Math expressions (examples):
after you review the keywords, test yourself

More vocabulary and key words:

· "Per" means "divided by"
as "I drove 90 miles on three gallons of gas, so I got 30 miles per gallon."
(Also 30 miles/gallon)

· "a" sometimes means "divided by"
as in "When I filled up, I paid $10.50 for three gallons of gasoline,
so the gas was 3.50 a gallon, or $3.50/gallon

· "less than"
If you need to translate "1.5 less than x", the temptation is to write "1.5 - x". DON'T! Put a "real world" situation in, and you'll see how this is wrong: "He makes $1.50 an hour less than me." You do NOT figure his wage by subtracting your wage from $1.50.
Instead, you subtract $1.50 from your wage

· "quotient/ratio of" constructions
If a problems says "the ratio of x and y",
it means "x divided by y"or x/y or x ÷ y

· "difference between/of" constructions
If the problem says "the difference of x and y",
it means "x - y"

In January of the year 2000, I was one more than eleven times as old as my son William. In January of 2009, I was seven more than three times as old as him. How old was my son in January of 2000?

Obviously, in "real life" you'd have walked up to my kid and and asked him how old he was, and he'd have proudly held up three grubby fingers, but that won't help you on your homework. Here's how you'd figure out his age for class:

First, name things and translate the English into math: Let "E " stand for my age in 2000, and let "W " stand for William's age. Then E = 11W + 1in the year 2000 (from "eleven times as much, plus another one"). In the year 2009 (nine years after the year 2000), William and I will each be nine years older, so our ages will be E + 9 and W + 9. Also, I was seven more than three times as old as William was, so E + 9 = 3(W + 9) + 7 = 3W + 27 + 7 = 3W + 34. This gives you two equations, each having two variables:

E = 11W + 1
E + 9 = 3W + 34

If you know how to solve systems of equations, you can proceed with those techniques. Otherwise, you can use the first equation to simplify the second: since E = 11W + 1, plug "11W + 1 " in for "E " in the second equation:

E + 9 = 3W + 34
(11W + 1) + 9 = 3W + 34
11W – 3W = 34 – 9 – 1
8W = 24
W = 3
Remember that the problem did not ask for the value of the variable W; it asked for the age of a person. So the answer is: William was three years old in January of 2000.

In three more years, Miguel's grandfather will be six times as old as Miguel was last year. When Miguel's present age is added to his grandfather's present age, the total is68. How old is each one now?

One-half of Heather's age two years from now plus one-third of her age three years ago is twenty years. How old is she now?

"Here lies Diophantus," the wonder behold . . .
Through art algebraic, the stone tells how old:
"God gave him his boyhood one-sixth of his life,
One twelfth more as youth while whiskers grew rife;
And then yet one-seventh ere marriage begun;
In five years there came a bouncing new son.
Alas, the dear child of master and sage
After attaining half the measure of his fathers life
chill fate took him.
After consoling his fate by this science of numbers
for four years, he ended his life."

Find Diophantus' age at death.

In order to solve geometric word problems, you will need to have memorized some geometric formulas for at least the basic shapes (circles, squares, right triangles, etc). You will usually need to figure out from the word problem which formula to use, and many times you will need more than one formula for one exercise. So make sure you have memorized any formulas that are used in the homework, because you may be expected to know them on the test.

Some problems are just straightforward applications of basic geometric formulae.

The radius of a circle is 3 centimeters. What is the circle's circumference?

The formula for the circumference C of a circle with radius r is:

C = 2(pi)r

...where "pi" (above) is of course the number approximately equal to 22 / 7 or 3.14159. They gave me the value of r and asked me for the value of C, so I'll just "plug-n-chug":

C = 2(pi)(3) = 6pi

Then, after re-checking the original exercise for the required units (so my answer will be complete):

the circumference is 6pi cm.

Note: Unless you are told to use one of the approximations for pi, or are told to round to some number of decimal places (from having used the "pi" button on your calculator), you are generally supposed to keep your answer in "exact" form, as shown above. If you're not sure if you should use the "pi" form or the decimal form, use both: "6pi cm, or about 18.85 cm".

A square has an area of sixteen square centimeters. What is the length of each of its sides?

The formula for the area A of a square with side-length s is:

A = s²

They gave me the area, so I'll plug this value into the area formula, and see where this leads:

16 = s²
4 = s

After re-reading the exercise to find the correct units, my answer is:

The length of each side is 4 centimeters.

A cube has a surface area of fifty-four square centimeters. What is the volume of the cube?

Work" problems involve situations such as two people working together to paint a house. You are usually told how long each person takes to paint a similarly-sized house, and you are asked how long it will take the two of them to paint the house when they work together. Many of these problems are not terribly realistic (since when do two laser printers work together on printing one report?), but it's the technique that they want you to learn, not the applicability to "real life".

The method of solution for work problems is not obvious, so don't feel bad if you're totally lost at the moment. There is a "trick" to doing work problems: you have to think of the problem in terms of how much each person / machine / whatever does in a given unit of time. For instance:

Suppose one painter can paint the entire house in twelve hours, and the second painter takes eight hours. How long would it take the two painters together to paint the house?

If the first painter can do the entire job in twelve hours and the second painter can do it in eight hours, then (this here is the trick!) the first guy can do ¹/₁₂ of the job per hour, and the second guy can do ¹/₈ per hour. How much then can they do per hour if they work together?

To find out how much they can do together per hour, I add together what they can do individually per hour: ¹/₁₂ + ¹/₈ = ⁵/₂₄. They can do ⁵/₂₄ of the job per hour. Now I'll let "t" stand for how long they take to do the job together. Then they can do ¹/_t per hour, so ⁵/₂₄ = ¹/_t. Flip the equation, and you get that t = ²⁴/₅ = 4.8 hours. That is:

hours to complete job:
first painter: 12
second painter: 8
together: t

completed per hour:
first painter: ¹/₁₂
second painter: ¹/₈
together: ¹/_t

adding their labor: Copyright © Elizabeth S1999-2011 All Rights Reserved

¹/₁₂ + ¹/₈ = ¹/_t

⁵/₂₄ = ¹/_t

²⁴/₅ = t

They can complete the job together in just under five hours.

One pipe can fill a pool 1.25 times faster than a second pipe. When both pipes are opened, they fill the pool in five hours. How long would it take to fill the pool if only the slower pipe is used?

Convert to rates:

hours to complete job:
fast pipe: f
slow pipe: 1.25f
together: 5

completed per hour:
fast pipe: ¹/_f
slow pipe: ¹/_1.25f
together: ¹/₅

adding their labor:

¹/_f + ¹/_1.25f = ¹/₅

multiplying through by 5f:

5 + 5/1.25 = f
5 + 4 = f = 9

Then 1.25f = 11.25, so the slower pipe takes 11.25 hours.

If you're not sure how I derived the rate for the slow pipe, think about it this way: If someone goes twice as fast as you, then you take twice as long as he does; if he goes three times as fast, then you take three times as long. In this case, if he goes 1.25 times as fast, then you take 1.25 times as long.

Two mechanics were working on your car. One can complete the given job in six hours, but the new guy takes eight hours. They worked together for the first two hours, but then the first guy left to help another mechanic on a different job. How long will it take the new guy to finish your car?

Working alone, Maria can complete a task in 100 minutes. Shaniqua can complete the same task in two hours. They work together for 30 minutes when Liu, the new employee, joins and begins helping. They finish the task 20 minutes later. How long would it take Liu to complete the task alone?

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