Translating Word Problems

The hardest thing about doing word problems is taking the English words and translating them into mathematics. Usually, once you get the math equation, you're fine; the actual math involved is often fairly simple. But figuring out the actual equation can seem nearly impossible. What follows is a list of hints and helps. Be advised, however: To really learn "how to do" word problems, you will need to practice, practice, practice.

The first step to effectively translating and solving word problems is to read the problem entirely. Don't start trying to solve anything when you've only read half a sentence. Try first to get a feel for the whole problem; try first to see what information you have, and what you still need.

The second step is to work in an organized manner. Figure out what you need but don't have, and name things. Pick variables to stand for the unknows, clearly labelling these variables with what they stand for. Draw and label pictures neatly. Explain your reasoning as you go along. And make sure you know just exactly what the problem is actually asking for. You need to do this for two reasons:

• Working clearly will help you think clearly, and
• figuring out what you need will help you translate your final answer back into English.

Regarding (2) above: It can be really frustrating (and embarassing) to spend fifteen minutes solving a word problem on a test, only to realize at the end that you no longer have any idea what "x" stands for, so you have to do the whole problem over again. I did this on a calculus test -- thank heavens it was a short test! -- and, trust me, you don't want to do this to yourself!

The third step is to look for "key" words. Certain words indicate certain mathematical operations. Below is a partial list.

Addition

• increased by
• more than
• combined, together
• total of
• sum
• added to

Subtraction

• decreased by
• minus, less
• difference between/of
• less than, fewer than

Multiplication

• of
• times,
• multiplied by
• product of
• increased/decreased by a factor of (this type can involve both addition or subtraction and multiplication!)

Division

• per, a
• out of
• ratio of, quotient of percent (divide by 100)

Equals

• is, are, was, were, will be gives, yields sold for

Note that "per" means "divided by", as in "I drove 90 miles on three gallons of gas, so I got 30 miles per gallon". Also, "a" sometimes means "divided by", as in "When I tanked up, I paid $12.36 for three gallons, so the gas was $4.12 a gallon".

Warning: The "less than" construction is backwards in the English from what it is in the math. If you need to translate "1.5 less than x", the temptation is to write "1.5 – x". Do not do this! You can see how this is wrong by using this construction in a "real world" situation: Consider the statement, "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. So remember; the "less than" construction is backwards.

Also note that order is important in the "quotient/ratio of" and "difference between/of" constructions. If a problems says "the ratio of x and y", it means "x divided by y", not "y divided by x". If the problem says "the difference of x and y", it means "x – y", not "y – x".

Examples

• Translate "the sum of 8 and y" into an algebraic expression. This translates to "8 + y"

• Translate "4 less than x" into an algebraic expression. This translates to "x – 4"

Remember? "Less than" is backwards in the math from how you say it in words!

• Translate "x multiplied by 13" into an algebraic expression. This translates to "13x"

• Translate "the quotient of x and 3" into an algebraic expression. This translates to " x/3"

• Translate "the difference of 5 and y" into an algebraic expression. This translates to "5 – y"

• Translate "the ratio of 9 more than x to x" into an algebraic expression. This translates to "(x + 9) / x"

• Translate "nine less than the total of a number and two" into an algebraic expression, and simplify. This translates to "(n + 2) – 9", which then simplifies to "n – 7"

Here are some more wordy examples:

• The length of a football field is 30 yards more than its width. Express the length of the field in terms of its width w.

Whatever the width w is, the length is 30 more than this. Recall that "more than" means "plus that much", so you'll be adding 30 to w. The expression they're looking for is "w + 30".

This one is important:

• Twenty gallons of crude oil were poured into two containers of different size. Express the amount of crude oil poured into the smaller container in terms of the amount g poured into the larger container.

The expression they're looking for is found by this reasoning: There are twenty gallons total, and we've already poured g gallons of it. How many gallons are left? There are 20 – g gallons left. They want the answer "20 – g".

This is the "how much is left" construction: You will be given some total amount. Smaller amounts, of unspecified sizes, are added (combined, mixed, etc) to create this total amount. You will pick a variable to stand for one of these unknown amounts. After having thus accounted for one of the amounts, the remaining amount is whatever is left after deducting this named amount from the total.

They may tell you that a trip took ten hours, and that the trip had two legs. You might name the time for the first leg as "t", with the remaining time for the second leg being 10 – t.

They may tell you that a hundred-pound order of animal feed was filled by mixing products from Bins A, B, and C, and that twice as much was added from Bin C as from Bin A. Let "a" stand for the amount from Bin A. Then the amount from Bin C was "2a", and the amount taken from Bin B was the remaining portion of the hundred pounds: 100 – a – 2a.

Reference:

http://www.purplemath.com/modules/translat2.htm

Rounding Numbers

When you have to round a number, you are usually told how to round it. It's simplest when you're told how many "places" to round to, but you should also know how to round to a named "place", such as "to the nearest thousand" or "to the ten-thousandths place". You may also need to know how to round to a certain number of significant digits; we'll get to that later.

In general, you round to a given place by looking at the digit one place to the right of the "target" place. If the digit is a five or greater, you round the target digit up by one. Otherwise, you leave the target as it is. Then you replace any digits to the right with zeroes (if they are to the left of the decimal point) or else you delete the digits (if they are past the decimal point).

I'll use the first few digits of the decimal expansion of pi: 3.14159265... in the examples below.

• Round pi to five places.

"To five places" means "to five decimal places". First, I count out the five decimal places, and then I look at the sixth place:

3.14159 | 265...

I've drawn a little line separating the fifth place from the sixth place. This can be a handy way of "keeping your place", especially if you are dealing with lots of digits.

The fifth place has a 9 in it. Looking at the sixth place, I see that it has a 2 in it. Since 2 is less than five, I won't round the 9 up; that is, I'll leave the 9 as it is. In addition, I will delete the digits after the 9. Then pi, rounded to five places, is:

3.14159

• Round pi to four places.

First, I go back to the original number (not the one I just rounded in the previous example). I count off four places, and look at the number in the fifth place:

3.1415 | 9265...

The number in the fifth place is a 9, which is greater than 5, so I'll round up in the fourth place, truncating the expansion at four decimal places. That is, the 5 becomes a 6, the 9265... part disappears, and pi, rounded to four decimal places, is:

3.1416

• Round pi to three places.

First, I go back to the original number (not the one I just rounded in the previous example). I count off three decimal places, and look at the digit in the fourth place:

3.141 | 59265...

The number in the fourth place is a 5, which is the cut-off for rounding: if the number in the next place (after the one you're rounding to) is 5 or greater, you round up. In this case, the 1 becomes a 2, the 59265... part disappears, and pi, rounded to three decimal places, is:

3.142

This rounding works the same way when they tell you to round to a certain named place, such as "the hundredths place". The only difference is that you have to be a bit more careful in counting off the places you need. Just remember that the decimal places count off to the right in the same order as the counting numbers count off to the left. That is, for regular numbers, you have the place values:

...(ten-thousands) (thousands) (hundreds) (tens) (ones)

For decimal places, you don't have a "oneths", but you do have the other fractions:

(decimal point) (tenths) (hundredths) (thousandths) (ten-thousandths)...

For instance:

• Round pi to the nearest thousandth.

"The nearest thousandth" means that I need to count off three decimal places (tenths, hundredths, thousandths), and then round:

3.141 | 59265...

Then pi, rounded to the nearest thousandth, is 3.142.

Round 2.796 to the hundredths place.
The hundredths place is two decimal places, so I'll count off two decimal places, and round according to the third decimal place:

2.79 | 6

Since the third decimal place contains a 6, which is greater than 5, I have to round up. But rounding up a 9 gives a 10. In this case, I round the 79 up to an 80:

2.80

Another consideration in rounding is when you have to round to "an appropriate number of significant digits". What are significant digits? Well, they're sort of the "interesting" or "important" digits. For example,

• 3.14159 has six significant digits (all the numbers give you useful information)

• 1000 has one significant digit (only the 1 is interesting; you don't know anything for sure about the hundreds, tens, or units places; the zeroes may just be placeholders; they may have rounded something off to get this value)

• 1000.0 has five significant digits (the ".0" tells us something interesting about the presumed accuracy of the measurement being made: that the measurement is accurate to the tenths place, but that there happen to be zero tenths)

• 0.00035 has two significant digits (only the 3 and 5 tell us something; the other zeroes are placeholders, only providing information about relative size)

• 0.000350 has three significant digits (that last zero tells us that the measurement was made accurate to that last digit, which just happened to have a value of zero)

• 1006 has four significant digits (the 1 and 6 are interesting, and we have to count the zeroes, because they're between the two interesting numbers)
• 560 has two significant digits (the last zero is just a placeholder)

• 560. (notice the "point" after the zero) has three significant digits (the decimal point tells us that the measurement was made to the nearest unit, so the zero is not just a placeholder)

• 560.0 has four significant digits (the zero in the tenths place means that the measurement was made accurate to the tenths place, and that there just happen to be zero tenths; the 5 and 6 give useful information, and the other zero is between significant digits, and must therefore also be counted)

If you need to express your answer as being "accurate to" a certain place, here's how the language works with the above examples: Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

• 3.14159 is accurate to the hundred-thousandths place
• 1000 is accurate to the thousands place
• 1000.0 is accurate to the tenths place
• 0.00035 is accurate to the hundred-thousandths place
• 0.000350 is accurate to the millionths place (note the extra zero)
• 1006 is accurate to the units place
• 560 is accurate to the tens place
• 560. is accurate to the units place (note the decimal point)
• 560.0 is accurate to the tenths place

Here are the basic rules for significant digits:

1) All nonzero digits are significant.
2) All zeroes between significant digits are significant.
3) All zeroes which are both to the right of the decimal point and to the right of all non-zero significant digits are themselves significant.

Here are some rounding examples; each number is rounded to four, three, and two significant digits.

Round 742,396 to four, three, and two significant digits:

• 742,400 (four significant digits)
• 742,000 (three significant digits)
• 740,000 (two significant digits)

Round 0.07284 to four, three, and two significant digits:

• 0.07284 (four significant digits)
• 0.0728 (three significant digits)
• 0.073 (two significant digits)

Round 231.45 to four, three, and two significant digits:
231.5 (four significant digits)
231 (three significant digits)
230 (two significant digits)

Rounding Addition

How do you round when they give you a bunch of numbers to add? You would add (or subtract) the numbers as usual, but then you would round the answer to the same decimal place as the least-accurate number.

Round to the appropriate number of significant digits:
13.214 + 234.6 + 7.0350 + 6.38

Looking at the numbers, I see that the second number, 234.6, is only accurate to the tenths place; all the other numbers are accurate to a greater number of decimal places. So my answer will have to be rounded to the tenths place:

13.214 + 234.6 + 7.0350 + 6.38 = 261.2290

Rounding to the tenths place, I get:
13.214 + 234.6 + 7.0350 + 6.38 = 261.2

Rounding Multiplication

How do you round, when they give you numbers to multiply (or divide)? You would multiply (or divide) the numbers as usual, but then you would round the answer to the same number of significant digits as the least-accurate number.

Simplify, and round to the appropriate number of significant digits:
16.235 × 0.217 × 5

First, I note that 5 has only one significant digit, so I will have to round my final answer to one significant digit. The product is:

16.235 × 0.217 × 5 = 17.614975

...but since I can only claim one accurate significant digit, I will need to round 17.614975 to 20, which is accurate to one significant digit.

16.235 × 0.217 × 5 = 20

How to Round Numbers

• Rule One. Determine what your rounding digit is and look to the right side of it. If the digit is 0, 1, 2, 3, or 4 do not change the rounding digit. All digits that are on the right hand side of the requested rounding digit will become 0.

• Rule Two. Determine what your rounding digit is and look to the right of it. If the digit is 5, 6, 7, 8, or 9, your rounding digit rounds up by one number. All digits that are on the right hand side of the requested rounding digit will become 0.

• Rounding with decimals: When rounding numbers involving decimals, there are 2 rules to remember:

Rule One Determine what your rounding digit is and look to the right side of it. If that digit is 4, 3, 2, or 1, simply drop all digits to the right of it.

Rule Two Determine what your rounding digit is and look to the right side of it. If that digit is 5, 6, 7, 8, or 9 add one to the rounding digit and drop all digits to the right of it.

Rule Three: Some teachers prefer this method:

This rule provides more accuracy and is sometimes referred to as the 'Banker's Rule'. When the first digit dropped is 5 and there are no digits following or the digits following are zeros, make the preceding digit even (i.e. round off to the nearest even digit). E.g., 2.315 and 2.325 are both 2.32 when rounded off to the nearest hundredth. Note: The rationale for the third rule is that approximately half of the time the number will be rounded up and the other half of the time it will be rounded down.

Reference:
http://www.purplemath.com/modules/rounding3.htm

http://math.about.com/od/arithmetic/a/Rounding.htm

Midpoint

The midpoint (also known as class mark in relation to histogram) is the middle point of a line segment. It is equidistant from both endpoints.

The midpoint of the segment (x1, y1) to (x2, y2)

Formulas
The formula for determining the midpoint of a segment in the plane, with endpoints (x1) and (x2) is:

The formula for determining the midpoint of a segment in the plane, with endpoints (x1, y1) and (x2, y2) is:

The formula for determining the midpoint of a segment in the plane, with endpoints (x1, y1, z1) and (x2, y2 z2) is:

More generally, for an n-dimensional space with axes the midpoint of an interval is given by:

Sometimes you need to find the point that is exactly between two other points. For instance, you might need to find a line that bisects (divides into equal halves) a given line segment. This middle point is called the "midpoint". The concept doesn't come up often, but the Formula is quite simple and obvious, so you should easily be able to remember it for later.

Think about it this way: If you are given two numbers, you can find the number exactly between them by averaging them, by adding them together and dividing by two. For example, the number exactly halfway between 5 and 10 is ^[5 + 10]/2 = 15/2 = 7.5.

Find the midpoint between (–1, 2) and (3, –6).
Apply the Midpoint Formula:

So the answer is P = (1, –2).

Find the midpoint between (6.4, 3) and (–10.7, 4).
Apply the Midpoint Formula:

So the answer is P = (–2.15, 3.5)

Find the value of p so that (–2, 2.5) is the midpoint between (p, 2) and (–1, 3).
I'll apply the Midpoint Formula:

This reduces to needing to figure out what p is, in order to make the x-values work:

So the answer is p = –3.

Reference:
http://en.wikipedia.org/wiki/Midpoint

http://www.purplemath.com/modules/midpoint.htm

Complex Numbers

A complex number is a number consisting of a real and imaginary part. It can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit with the property i^ 2 = −1 The complex numbers contain the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the understanding of addition and multiplication.

Complex numbers were first conceived and defined by the Italian mathematician Gerolamo Cardano, who called them "fictitious", during his attempts to find solutions to cubic equations.The solution of a general cubic equation in radicals (without trigonometric functions) may require intermediate calculations containing the square roots of negative numbers, even when the final solutions are real numbers, a situation known as casus irreducibilis. This ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed field, where any polynomial equation has a root.

A complex number can be visually represented as a pair of numbers forming a vector on a diagram called an Argand diagram, representing the complex plane.

Operations

Complex numbers are added, subtracted, multiplied, and divided by formally applying the associative, commutative and distributive laws of algebra, together with the equation i^ 2 = −1:

Addition:

Subtraction:

Multiplication:


Division:

where c and d are not both zero. This is obtained by multiplying both the numerator and the denominator by the conjugate of the denominator c + di, which is (c − di).

Reference:
http://en.wikipedia.org/wiki/Complex_number

Polynomials

By now, you should be familiar with variables and exponents, and you may have dealt with expressions like 3x4 or 6x. Polynomials are sums of these "variables and exponents" expressions. Each piece of the polynomial, each part that is being added, is called a "term". Polynomial terms have variables which are raised to whole-number exponents (or else the terms are just plain numbers); there are no square roots of variables, no fractional powers, and no variables in the denominator of any fractions.
Here is a typical polynomial:


Notice the exponents on the terms. The first term has an exponent of 2; the second term has an "understood" exponent of 1; and the last term doesn't have any variable at all. Polynomials are usually written this way, with the terms written in "decreasing" order; that is, with the largest exponent first, the next highest next, and so forth, until you get down to the plain old number.

Any term that doesn't have a variable in it is called a "constant" term because, no matter what value you may put in for the variable x, that constant term will never change. In the picture above, no matter what x might be, 7 will always be just 7.

The first term in the polynomial, when it is written in decreasing order, is also the term with the biggest exponent, and is called the "leading term".
The exponent on a term tells you the "degree" of the term. For instance, the leading term in the above polynomial is a "second-degree term" or "a term of degree two". The second term is a "first degree" term. The degree of the leading term tells you the degree of the whole polynomial; the polynomial above is a "second-degree polynomial". Here are a couple more examples:

Give the degree of the following polynomial: 2x^5 – 5x^3 – 10x + 9

This polynomial has four terms, including a fifth-degree term, a third-degree term, a first-degree term, and a constant term.

This is a fifth-degree polynomial.

Give the degree of the following polynomial: 7x^4 + 6x^2 + x

This polynomial has three terms, including a fourth-degree term, a second-degree term, and a first-degree term. There is no constant term.

This is a fourth-degree polynomial.

When a term contains both a number and a variable part, the number part is called the "coefficient". The coefficient on the leading term is called the "leading" coefficient.

In the above example, the coefficient of the leading term is 4; the coefficient of the second term is 3; the constant term doesn't have a coefficient.

The "poly" in "polynomial" means "many". I suppose, technically, the term "polynomial" should only refer to sums of many terms, but the term is used to refer to anything from one term to the sum of a zillion terms. However, the shorter polynomials do have their own names:

1. a one-term polynomial, such as 2x or 4x2, may also be called a "monomial" ("mono" meaning "one")
2. a two-term polynomial, such as 2x + y or x2 – 4, may also be called a "binomial" ("bi" meaning "two")
3.a three-term polynomial, such as 2x + y + z or x4 + 4x2 – 4, may also be called a "trinomial" ("tri" meaning "three")

Polynomials are also sometimes named for their degree:

1.a second-degree polynomial, such as 4x2, x2 – 9, or ax2 + bx + c, is also called a "quadratic"
2.a third-degree polynomial, such as –6x3 or x3 – 27, is also called a "cubic"
3.a fourth-degree polynomial, such as x4 or 2x4 – 3x2 + 9, is sometimes called a "quartic"
4.a fifth-degree polynomial, such as 2x5 or x5 – 4x3 – x + 7, is sometimes called a "quintic"

"Evaluating" a polynomial is the same as evaluating anything else: you plug in the given value of x, and figure out what y is supposed to be. For instance:

Evaluate 2x^3 – x^2 – 4x + 2 at x = –3

I need to plug in "–3" for the "x", remembering to be careful with my parentheses and the negatives:

2(–3)3 – (–3)2 – 4(–3) + 2
= 2(–27) – (9) + 12 + 2
= –54 – 9 + 14
= –63 + 14
= –49

Always remember to be careful with the minus signs!


Reference:
http://www.purplemath.com/modules/polydefs2.htm