Overview
Multiplication of fractions and rates with variables is similar to multiplying regular fractions and rates. It is important to cancel out common factors and units.
The General Pattern
The general pattern for multiplying fractions is to multiply the numerators and multiply the denominators to create a new fraction. In symbol form, the rule is for real numbers a, b, c, and d, where b and d are not equal to zero, a/b ∙c/d is equal to ac/bd. Following the pattern, 2/3 ∙7x/4 can be solved by (2∙7x)/(3∙4), or 14x/12. In simplest terms, the new fraction is 7x/6, as both the numerator and denominator can be divided by their common factor 2/2.
Figure 1: The rule in symbol form for multiplying fractions.
Factoring Before Multiplying
Another way to solve a problem where there are common factors is to take out the common factors before multiplying. A problem such as (2∙ 7x)/(3∙4) has common factors that can be cancelled out before it is multiplied. Using the Commutative Property, the problem becomes (2/4)(7x/3). Using Equal Fractions, 2/4 can be simplified to ½ and then multiplied as 7x/(2∙3), or 7x/6. Suppose the problem is 9/(3y) ∙6y. Another way to simplify the problem is to set it up as multiplication of the fractions 9/(3y) ∙6y/1, so it looks like (9∙6y)/(3y). That simplifies even further, because 6y is evenly divisible by 3y, as 2.
Figure 2: The process of factoring before multiplying.
Rates
Rates can be multiplied by other quantities. Suppose a car is traveling at a rate of 70 miles per hour. In 2 hours, it will travel 140 miles. The hours in the miles per hour and the time traveled (2 hours) cancel each other out. Similarly, suppose a typist types 70 words per minute. In 10 minutes, the typist can type a 700-word document.
Multiplying Rates
Rates can be multiplied similar to fractions. Suppose a commuter drives 30 minutes per day to work, 5 days per week. That commuter drives 30 min/day times 5 days/week. How many minutes per week does she drive? The days cancel out, so the equation becomes 30 minutes times 5 or 150 minutes/week. Also, 150 minutes/week can be simplified to 2 hours and 30 minutes, by dividing by 60 minutes/ hour.
Figure 3: Rates can be multiplied. For example, commuting time is equal to time per day multiplied by days per week.
Interested in algebra tutoring services? Learn more about how we are assisting thousands of students each academic year.
SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Huron, SD: visit Tutoring in Huron, SD