Download Multiplying Fractions

Multiplying Fractions
1. First one must convert all mixed numbers and whole numbers into improper
fractions.
a. To convert whole numbers into an improper fraction put the whole number
over one.
Example 1: 4 = 4/1
Example 2: 25 = 25/1
b. To convert mixed numbers into improper fractions take the whole number
times the denominator then add the numerator to that product and put that
answer over the original denominator.
Example 1: 4 2/3 = [(4 x 3) + 2]/3 = 14/3
Example 2: 3 ¼ = [(3 x 4) + 1]/3 = 13/3
2. Then multiply the numerators (top numbers) together, then multiply the
denominators (bottom numbers) together. Put the product of the numerators
over the product of the denominators
Example 1: 4/5 x 3/7 = (4 x 3) / (5 x 7) = 12/35
Example 2: 26/5 x 17/13 = (26 x 17) / (5 x 13) = 442/65
3. Then check to see if the answer is in simplest form.
a. Reducing a fraction: Find the greatest common factor (GCF) of the
numerator and denominator. Then divide both the numerator and
denominator by the GCF. Then the fraction is reduced. If the GCF of the
numerator and denominator is one, then the fraction is in simplest form.
Example 1: 24/54 : GCF(24,54) = 6 thus (24/6) / (54/6) = 4/9
Example 2: 442/65 : GCF(442,65) = 13 : (442/13) / (65/13) = 34/5
b. Cross-Canceling: Divide the numerator of the first fraction and the
denominator of the second fraction by the GCF of them. Cross them out
and put the quotient of each of them in their place. Then do the same with
the denominator of the first fraction with the numerator of the second
fraction. Finally multiply the fractions with the new numerators and
denominators to get a reduced fraction as the final answer.
Example 1: 4/5 x 15/32 = (4/4) / (5/5) x (15/5) / (32/4) = 1/1 x 3/8 = 3/8
Example 2: 4/9 x 3/14 = (4/2)/(9/3) x (3/3)/(14/2) = 2/3 x 1/7 = 2/21
4. Finally, one can convert improper fractions into mixed numbers. Divide the
numerator by the denominator to find the whole number, then take the
remainder and put it back over the denominator to make a mixed number.
Example 1: 34/5 = 5 goes into 34 6 times with 4 left over: 6 4/5
Example 2: 23/4 = 4 goes into 23 5 times with 3 left over: 5 ¾
Dividing Fractions
1. Convert all whole numbers and mixed numbers into improper fractions. (see
multiplying fractions 1a and 1b)
2. Keep the first fraction the same, change the division sign to a multiplication
sign and use the reciprocal (multiplicative inverse) of the second fraction (just
flip the fraction).
Example 1: (2/3) / (5/7) = (2/3) x (7/5) = 14/15
Example 2: (1 ½) / (2 1/3) = (3/2) / (7/3) = (3/2) x (3/7) = 9/14
3. Finally, follow the rules for multiplying fractions.
Adding Fractions
1. One must have common denominators to add fractions or mixed numbers.
 Finding Common Denominators: Find the least common multiple (LCM) of
the denominators. Then convert each fraction into an equivalent fraction
using the LCM as the new denominator of each fraction.
Example 1: 2/3 + ¼ = (2/3 x 4/4) + (1/4 x 3/3) = 8/12 + 3/12
Example 2: ¾ + 1/8 = (3/4 x 2/2) + (1/8 x 1/1) = 6/8 + 7/8
2. After getting common denominators, add the numerators (top numbers)
together and put the sum of the numerators over the common denominator of
the fractions.
Example 1: 8/12 + 3/12 = (8 + 3) / 12 = 11/12
Example 2: 6/8 + 7/8 = (6 + 7) / 8 = 13/8
3. In the case of whole number and a fraction, put the whole number in front of
the fraction to create a mixed number.
Example 1: 3 + 1/6 = 3 1/6
Example 2: 7 + 4/5 = 7 4/5
4. In the case of a whole number and a mixed number, add the whole numbers
together and then add the fractional part of the mixed number to it to make a
new mixed number.
Example 1: 4 + 3 2/3 = (4 + 3) 2/3 = 7 2/3
Example 2: 12 + 7 5/6 = (12 + 7) 5/6 = 19 5/6
5. In the case of two mixed numbers, add the two whole numbers together and
then add the two fractional parts together and put the two sums together to
create a new mixed number.
Example 1: 2 1/5 + 3 3/5 = (2 + 3) and (1/5 + 3/5) = 5 and 4/5 = 5 4/5
Example 2: 3 3/5 + 6 4/5 = (3 + 6) and (3/5 + 4/5) = 9 and 7/5 = 9 7/5
 A mixed number cannot have an improper fraction as its fractional part of
the mixed number. Convert the improper fraction into a mixed number and
add that mixed number to the whole number part of the mixed number.
Example 1: 9 7/5 = 9 + 1 2/5 = 10 2/5
Example 2: 2 14/9 = 2 + 1 5/9 = 3 5/9
6. Finally, check to see if the fractional part of the answer is in simplest form.
7. Simplest form: To check to see if a fraction is in simplest form, find the
greatest common factor (GCF) of the numerator and denominator. If the GCF
of the numerator and denominator is one, then the fraction is in simplest form.
If the GCF is some other number, then divide the numerator by the GCF to get
a new numerator, then divide the denominator by the GCF to get a new
denominator. Finally, put the new numerator over the new denominator to get
a new fraction in simplest form. In the case of mixed numbers, simplify the
fractional part of the mixed number and keep the whole number the same.
Example 1: 3 2/4 = 3 (2/2)/(4/2) = 3 1/2
Example 2: 13 24/36 = 13 (24/12)/ (36/12) = 13 2/3
8. Converting: When the numerator of a fraction is equal to or greater than the
denominator, one may need to convert it to a whole number or mixed number.
To convert, divide the numerator by the denominator and put the dividend as
the answer. The dividend will be a whole number if the denominator divides
into the numerator with no remainder or it is a mixed number with the
remainder going over the denominator to make the fractional part of the mixed
number.
Example 1: 8/4 = (8/4) = 2 Example 2: 2 3/9 = 2 (3/3)/(9/3) = 2 1/3
Subtracting Fractions
1. One must have common denominators to subtract fractions or mixed numbers.
(see finding common denominators under adding fractions).
2. After getting common denominators, subtract the numerators (top numbers) of
the fractions and put the difference over the common denominator.
Example 1: 7/8 – 3/8 = (7 – 3)/8 = 4/8
Example 2: 17/25 – 13/25 = (17 – 13)/25 = 4/25
3. Whole number and a fraction: One must borrow from the whole number. One
is borrowing one whole, so one writes it as a number over itself. The number
must be the same as the denominator of the other fraction.
Example 1: 1 – 3/5 = 5/5 – 3/5 = 2/5
Example 2: 3 – 4/7 = 2 7/7 – 4/7 = 2 and (7/7 –4/7) = 2 and 3/7 = 2 3/7
4. Borrowing with mixed numbers: Borrow from the whole number part of the
mixed number and add it to the fractional part of the mixed number. The short
cut is to borrow from the whole number part, then just add the numerator and
denominator together and put that sum (new numerator) over the original
denominator to subtract from.
Example 1: 3 1/7 – 1 4/7 = 2 8/7 – 1 4/7 = (2 – 1) & (8/7 – 4/7) = 1 4/7
Example 2: 7 1/13 – 2 8/13 = 6 14/13 – 2 8/13 =
(6 – 2) and (14/13 – 8/13) = 4 and 6/13 = 4 6/13
5. Finally check to see if the fractional part is in simplest form.