Algebra I Workbook For Dummies
Page 4
Converting Improper and Mixed Fractions
An improper fraction is one where the numerator (the number on the top of the fraction) has a value greater than or equal to the denominator (the number on the bottom of the fraction) — the fraction is top heavy. Improper fractions can be written as mixed numbers or whole numbers — and vice versa. For example, is an improper fraction, and is a mixed number.
To change an improper fraction to a mixed number, divide the numerator by the denominator and write the remainder in the numerator of the new fraction.
To change a mixed number to an improper fraction, multiply the whole number times the denominator and add the numerator. This result goes in the numerator of a fraction that has the original denominator still in the denominator.
Q. Change to a mixed number.
A. . First, divide the 29 by 8:
Then write the mixed number with the quotient (the number of times the denominator divides into the numerator) as the whole number and the remainder as the numerator of the fraction in the mixed number: .
Q. Change to an improper fraction: .
A. . Multiply the 6 and 7 and then add the 5, which equals 47. Then write the fraction with this result in the numerator and the 7 in the denominator: .
1. Change the mixed number to an improper fraction.
Solve It
2. Change the mixed number to an improper fraction.
Solve It
3. Change the improper fraction to a mixed number.
Solve It
4. Change the improper fraction to a mixed number.
Solve It
Finding Fraction Equivalences
In algebra, all sorts of computations and manipulations use fractions. In many problems, you have to change the fractions so that they have the same denominator or so that their form is compatible with what you need to solve the problem. Two fractions are equivalent if they have the same value, such as and . To create an equivalent fraction from a given fraction, you multiply or divide the numerator and denominator by the same number. This technique is basically the same one you use to reduce a fraction.
Q. Find a fraction equivalent to with a denominator of 40.
A. . Because 5 times 8 is 40, you multiply both the numerator and denominator by 5. In reality, you’re just multiplying by 1, which doesn’t change the real value of anything.
Q. Reduce by multiplying the numerator and denominator by . The same thing is accomplished if you divide both numerator and denominator by 3.
A.
or
5. Find an equivalent fraction with a denominator of 28 for .
Solve It
6. Find an equivalent fraction with a denominator of 30 for .
Solve It
7. Reduce this fraction: .
Solve It
8. Reduce this fraction: .
Solve It
Making Proportional Statements
A proportion is an equation with two fractions equal to one another. Proportions have some wonderful properties that make them useful for solving problems — especially when you’re comparing one quantity to another or one percentage to another.
Given the proportion , then the following are also true:
a × d = c × b (The cross products form an equation.)
(The “flip” is an equation.)
(You can reduce either fraction vertically.)
(You can reduce the numerator or denominator horizontally.)
Q. Find the missing value in the following proportion:
A. 44. The numerator and denominator in the fraction on the left have a common factor of 6. Multiply each by . Flip the proportion to get the unknown in the numerator of the right-hand fraction. Then you see that the two bottom numbers each have a common factor of 7. Divide each by 7. Finally, cross-multiply to get your answer:
Q. If Agnes can type 60 words per minute, then how long will it take her to type a manuscript containing 4,020 words (if she can keep typing at the same rate)?
A. 67 minutes (1 hour and 7 minutes). Set up a proportion with words in the two numerators and the corresponding number of minutes in the denominators:
Divide both numerators by 60 and then cross-multiply to solve for x.
9. Solve for x:
Solve It
10. Solve for x:
Solve It
11. Solve for x:
Solve It
12. Solve for x:
Solve It
13. A recipe calls for 2 teaspoons of cinnamon and 4 cups of flour. You need to increase the flour to 6 cups. To keep the ingredients proportional, how many teaspoons of cinnamon should you use?
Solve It
14. A factory produces two faulty iPods for every 500 iPods it produces. How many faulty iPods would you expect to find in a shipment of 1,250?
Solve It
Finding Common Denominators
Before you can add or subtract fractions, you need to find a common denominator for them. Ideally that common denominator is the least common multiple — the smallest number that each of the denominators can divide into without a remainder. A method of last resort, though, is to multiply the denominators together. Doing so gives you a number that the denominators divide evenly. You may have to work with larger numbers using this method, but you can always reduce the fractions at the end.
Q. How would you write the fractions and with the same denominator?
A. and . The fractions and have denominators with no factors in common, so the least common denominator is 12, the product of the two numbers. Now you can write them both as fractions with a denominator of 12:
and
Q. What is the least common denominator for the fractions and
A. 120. The fractions and have denominators with a greatest common factor of 4. So multiplying the two denominators together gives you 480, and then dividing 480 by that common factor gives you 480 ÷ 4 = 120. The least common denominator is 120.
15. Rewrite the fractions and with a common denominator.
Solve It
16. Rewrite the fractions and with a common denominator.
Solve It
17. Rewrite the fractions and with a common denominator.
Solve It
18. Rewrite the fractions and with a common denominator.
Solve It
19. Rewrite the fractions , , and with a common denominator.
Solve It
20. Rewrite the fractions , , and with a common denominator.
Solve It
Adding and Subtracting Fractions
You can add fractions together or subtract one from another if they have a common denominator. After you find the common denominator and change the fractions to their equivalents, then you can add the numerators together or subtract them (keeping the denominators the same).
Q.
A. . First find the common denominator, 24, and then complete the addition:
Q. + + =
A. . You need a common denominator of 30:
The whole number parts are separated from the fractional parts to keep the numbers in the computations smaller.
Q. –
A. . In this problem, change both mixed numbers to improper fractions. The common denominator is 56:
.
Q. 1 –
A. . Even though the 1 isn’t a fraction, you need to write it as a fraction with a denominator of 13. The subtraction problem becomes .
21.
Solve It
22.
Solve It
23.
Solve It
24.
Solve It
Multiplying and Dividing Fractions
Multiplying fractions is really a much easier process than adding or subtracting fractions, because you don’t have to find a common denominator. Furthermore, you can take some creative steps and reduce the fractions before you even multiply them.
When multiplying fractions, you can pair up the numerator of any fraction in the problem with the d
enominator of any other fraction; then divide each by the same number (reduce). Doing so saves your having large numbers to multiply and then to reduce later.
When you start with mixed numbers, you have to change them to improper fractions before starting the reduction and multiplication process.
Algebra really doesn’t have a way to divide fractions. If you want to divide fractions, you just have to change them to multiplication problems. Sounds easy, right? Just change the division to multiplication and use the reciprocal — where the numerator and denominator switch places — of the second fraction in that new problem. The answer to this multiplication problem is the same as the answer to the original division problem.
Q.
A. . First, reduce: The 25 and 30 have a common factor of 5, the 14 and 49 have a common factor of 7, the 6 and 27 have a common factor of 3, and the 5 and 10 have a common factor of 5.
Reduce the 5 and 10 by dividing by 5. And then, to multiply the fractions, multiply all the numerators together and all the denominators together to make the new fraction:
Q.
A. .
First, write the mixed numbers as improper fractions. Then reduce where possible and multiply.
Q.
A. . First change the mixed numbers to improper fractions. Then change the divide to multiply and the second (right) fraction to its reciprocal. Finally, do the multiplication problem to get the answer:
Q. 2 ÷
A. . First change the 2 to a fraction: .
Then change the divide to multiply and the second (right) fraction to its reciprocal. Then do the multiplication problem to get the answer , which can be changed to the mixed number.
25.
Solve It
26.
Solve It
27.
Solve It
28.
Solve It
29.
Solve It
30.
Solve It
Simplifying Complex Fractions
A complex fraction is a fraction within a fraction. If a fraction has another fraction in its numerator or denominator (or both), then it’s called complex. Fractions with this structure are awkward to deal with and need to be simplified. To simplify a complex fraction, you first work at creating improper fractions or integers in the numerator and denominator, independently, and then you divide the numerator by the denominator.
Q.
A. . First, change the mixed number in the numerator to an improper fraction. Then divide the two fractions by multiplying the numerator by the reciprocal of the denominator.
Q.
A. . First, find a common denominator for the fractions in the numerator separate from those in the denominator. Then subtract the fractions in the numerator and add the fractions in the denominator. Finally divide the two fractions by multiplying the numerator by the reciprocal of the denominator.
31.
Solve It
32.
Solve It
33.
Solve It
34.
Solve It
Changing Fractions to Decimals and Vice Versa
Every fraction with an integer in the numerator and denominator has a decimal equivalent. Sometimes these decimals end (terminate); sometimes the decimals go on forever, repeating a pattern of digits over and over.
To change a fraction to a decimal, divide the denominator into the numerator, adding zeros after the decimal point until the division problem either ends or shows a repeating pattern. To indicate a pattern repeating over and over, draw a line across the top of the digits that repeat (for example, ) or just write a few sets of repeating digits (such as, 0.2345345345…) and put dots at the end.
To change a decimal to a fraction, place the digits of a terminating decimal over a power of 10 with as many zeros as there are decimal values. Then reduce the fraction. To change a repeating decimal to a fraction (this tip works only for those repeating decimals where the same digits repeat over and over from the beginning — with no other digits appearing), place the repeating digits in the numerator of a fraction that has 9’s in the denominator. It should have as many 9’s as digits that repeat. For instance, in the repeating decimal .123123123 . . . you’d put 123 over 999 and then reduce the fraction.
Q. Change to a decimal.
A. 0.04. Divide 25 into 1 by putting a decimal after the 1 and adding two zeros.
Q. Change to a decimal.
A. or 0.454545… Divide 11 into 5, adding zeros after the decimal, until you see the pattern.
Q. Change 0.452 to a fraction.
A. . Put 452 over 1,000 and reduce.
Q. Change to a fraction.
A. . Put 285714 over 999,999 and reduce.
The divisors used to reduce the fractions are, in order, 9, 3, 11, 13, and 37. Whew!
35. Change to a decimal.
Solve It
36. Change to a decimal.
Solve It
37. Change to a decimal.
Solve It
38. Change 0.45 to a fraction.
Solve It
39. Change to a fraction.
Solve It
40. Change to a fraction.
Performing Operations with Decimals
Decimals are essentially fractions whose denominators are powers of 10. This property makes for much easier work when adding, subtracting, multiplying, or dividing.
When adding or subtracting decimal numbers, just line up the decimal points and fill in zeros, if necessary.
When multiplying decimals, just ignore the decimal points until you’re almost finished. Count the number of digits to the right of the decimal point in each multiplier, and the total number of digits is how many decimal places you should have in your answer.
Dividing has you place the decimal point first, not last. Make your divisor a whole number by moving the decimal point to the right. Then adjust the number you’re dividing into by moving the decimal point the same number of places. Put the decimal point in your answer directly above the decimal point in the number you’re dividing into (the dividend).
Q.
A. 12.236004. Line up the decimal points in the first two numbers and add. Put in zeroes to help you line up the digits. Then subtract the last number from the result.
Q.
A. 0.191333…. Multiply the first two numbers together, creating an answer with four decimal places to the right of the decimal point. Then divide the result by 36, after moving the decimal point one place to the right in both divisor and dividend.
41.
Solve It
42.
Solve It
Answers to Problems on Fractions
This section provides the answers (in bold) to the practice problems in this chapter.
1. Change the mixed number to an improper fraction. The answer is .