Fraction Word Problems Study Guide (page 3)
Introduction to Fraction Word Problems
Arithmetic is numbers you squeeze from your head to your hand to your pencil to your paper till you get the answer.
—CARL SANDBURG (1878–1967)
This lesson will provide practice performing operations with fractions, as well as strategies that can be used when you are solving word problems with fractions.
If you look up the definition of the set of rational numbers, you may get a description like the one mentioned in the previous lesson. They are the set of numbers that can be expressed as , where b is not equal to zero, and a and b are both integers. This is just a complicated description of very familiar numbers known as fractions. Before embarking on our study of fractions and word problems involving fractions, let's review a few key concepts.
Factors, Multiples, GCFs, and LCMs
A Factor is a number that divides into another number without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
Greatest Common Factor (GCF)
The Greatest Common Factor, or GCF, is the largest value that divides each of the terms without a remainder.
- Example: Find the greatest common factor of 18 and 24.
- The factors of 18 are 1, 2, 3, 6, 9, and 18.
- The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
- The factors 1, 2, 3, and 6 are common to both lists, but the greatest common factor is 6.
A Multiple is the result of multiplying a number by another whole number. For example, multiples of 4 are 4 × 1 = 4, 4 × 2 = 8, 4 × 3 = 12, and so on.
Least Common Multiple (LCM)
The Least Common Multiple, or LCM, is the smallest value that each term divides into without leaving a remar.
- Example: Find the least common multiple of 12 and 30.
- Multiples of 12 are 12, 24, 36, 48, 60, 72 …
- Multiples of 30 are 30, 60, 90 …
- The smallest common number in each list is 60, so 60 is the least common multiple of 12 and 30.
To help remember the difference between factors and multiples, think factors fit into a number, and you multiply to get multiples.
We learn to simplify fractions in order to make them easier to use. If the numbers are smaller, they are usually less difficult to use.
To simplify a fraction, divide the numerator (top) and the denominator (bottom) by the greatest common factor. For example, take the fraction . The greatest common factor of the numerator and the denominator is 4. Divide each value by 4 to simplify the fraction to its lowest terms: .
An improper fraction has a numerator whose absolute value is greater than or equal to the absolute value of its denominator: is an improper fraction.
A mixed number is a number made up of a whole number part and a fraction part. The number is a mixed number.
Changing Improper Fractions and Mixed Numbers
Changing Improper Fractions to Mixed Numbers
To change an improper fraction into a mixed number, divide the numerator by the denominator. Then, place the remainder, if there is one, over the same denominator. For example, to change to a mixed number, divide 12 by 7. The result is 1, with a remainder of 5. The mixed number is .
Changing Mixed Numbers to Improper Fractions
To change a mixed number into an improper fraction, multiply the whole number part by the denominator. Then, add this product to the numerator. This value is the new numerator, and the denominator remains the same. For example, to change the mixed number to an improper fraction, multiply 4 by 3 to get 12. Then, add 12 + 1 = 13 to get the new numerator. Because the denominator remains the same, the improper fraction is .
Adding and Subtracting Fractions
To add or subtract fractions, you must ﬁrst have a common denominator. Then, add or subtract the numerators and keep the denominators. For example, in order to add , add the numerators and keep the denominator of 7. The result is .
On the other hand, to subtract , use the least common multiple of the denominators to ﬁnd the least common denominator. The least common multiple of 4 and 2 is 4, so change the second fraction to also have a denominator of 4. To do this, multiply the numerator and denominator by 2:. Now that there is a common denominator, subtract the numerators and keep the denominator..
When you are adding or subtracting mixed numbers, add or subtract the fraction parts and add or subtract the whole number parts. For example, to add , the fraction part adds to and the whole number parts add to 2 + 1 = 3. The answer is .
Use the greatest common factor (GCF) to simplify fractions, and use the least common multiple (LCM) to find a common denominator.
To multiply fractions, multiply across the numerators and denominators. Then, simplify the product if necessary. For example, . Since 12 and 45 share a greatest common factor of 3, divide each by 3 to simplify the fraction. The simpliﬁed fraction is .
If the numerator of a fraction has a common factor with the denominator of a fraction in a multiplication problem, you may cross cancel the common factors. For example, when multiplying , 3 and 9 share a common factor of 3. By canceling out this factor, the problem becomes . The fraction is in simplest form.
To multiply mixed numbers, ﬁrst change to improper fraction form, and then follow the same steps as before. For example, .
Dividing fractions is very similar to multiplying fractions. When you are dividing, change the problem so that you are multiplying by the reciprocal of the divisor. Then, multiply the fractions as usual. For example, in order to divide , ﬁrst change the problem to multiplication by multiplying by the reciprocal. The reciprocal of a fraction switches the numerator and the denominator. For example, the reciprocal of is . The problem becomes . Now cross cancel the common factors and multiply across: . As in multiplication, when you are dividing mixed numbers, change them to improper fraction form ﬁrst and then follow the same steps as before.
Fraction Word Problems
Now it is time to apply this knowledge of fractions and operations with fractions to math word problems. Use the following sample question as an example, and then try the practice questions that follow to test your skills.
Jamie needs yards of material for her school project. If she will bring in enough material for herself and three classmates, how much material does she need altogether?
Read and understand the question. Jamie needs material for herself and three others, so she needs four times the number of yards for one person.
Make a plan. Use key words to help solve this problem. The key word altogether in this context suggests multiplication. Multiply the number of yards for one person by 4 to find the total amount.
Carry out the plan. = 3. Jamie needs a total of 3 yards of material for herself and three classmates.
Check your answer. One way to check the result is to divide the total amount needed by the number of people. Three yards divided by 4 people = 3 ÷ 4 = yard per person. This solution is checking.
Find practice problems and solutions for these concepts at Fraction Word Problems Practice Questions.
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