Solving Proportions with a Variable Term

Size: px

Start display at page:

Download "Solving Proportions with a Variable Term"

Sharleen Powers
7 years ago
Views:

1 Section 2 : Proportions A proportion is an equation that states that two ratios or rates each written as fractions are equal to each other. A proportion is made by writing an equation with two proper or improper fractions set equal to each other. We do not use mied numbers when we write a proportion. It is common for the fractions in the proportion to have numbers that have units of measure. If that happens then the units of measure must be included when you write the proportion. 4 = = 24 2 dogs 5 dogs = 6 dogs 15 dogs 0 miles 2 miles = 120 miles miles states that the two cross products can be set equal to each other to form a new equation. Given the Proportion: A B A B = C D A D and B C and C D A D = B C Solving Proportions with a Variable Term If one of the terms in a proportion is an unknown variable then we can find the value for that variable term by using the Cross Product Rule for Proportions. Use the Cross Product Rule to create a simple one step equation and then solve that equation for. Given the Proportion: = and 5 and 20 = 5 10 Given the Proportion: = and 10 and 15 = 0 Math 100 Section 2 Page Eitel 10 15

2 Solving a Proportion for an Unknown If one of the numbers in a proportion is unknown then we can find the value for that number by using the Cross Product Rule for proportions. 1. Set the cross products equal to each other. 2. Solve the equation for.. Check your answer. Eample 1 Eample 2 4 = and 12 2 = and = 12 4 = 6 = 2 15 = 0 divide both sides by 4 4 / 4 / = 6 4 = 9 divide both sides by / / = 0 =10 Eample Eample = 7 2 = = 5 divide both sideds by 2 = 5 2 = = = = 28 divide both sideds by 2 =14 Math 100 Section 2 Page Eitel

3 Solving Proportions Word Problems Involving Rates There are many real world problems that involve proportions. The most common ones involve rates. A rate is a ratio of two numbers each with a different unit. The first rate is compared to a second rate but one of the numbers is unknown and its value must be found. To solve these problems we set up a proportion and then solve for the unknown. It is important to keep the units attached to the number values. You must also be sure that the numerators of the proportion have numbers with a common unit and the denominators also have numbers with a common unit. Eample 1 : A hospital requires a staffing ratio of three nurses for every two doctors. If the hospital employs 24 doctors, how many nurses are employed? Step 1. Find the sentence that compares two given values as a ratio. Write that ratio as a fraction and make sure that the units are attached to the values. nurses 2 doctors Step 2. Find the sentence that asks you to find a missing value given a known value. Write a fraction with the known and unknown with the units they come with. Make sure that this fraction has the units in the same position as the first fraction. nurses 24 doctors Step. Set the fraction in Step 1 equal to the fraction in Step 2. nurses 2 doctors = nurses 24 doctors Step 4. Set the cross products equal to each other and solve for the unknown value. Be sure to include units in the answer. nurses nurses 2 doctors 24 doctors 2 and 24 2 = 24 2 = 72 = 6 nurses Step 5. Check your answer. Check: 6 nurses 24 doctors = / 6 / 2 / 4 / 2 = nurses 2 doctors Math 100 Section 2 Page 2015 Eitel

4 Eample 2 : An engine requires a ratio of 5 ounces of oil for every 6 gallons of gas. If we need 0 gallons of gas how much oil should we add? The ratio of oil to gas is 5 ounces to 6 gallons 5 ounces 6 gallons We have 0 gallons for an unknown number of ounces 5 ounces 6 gallons = ounces 0 gallons 6 = =150 = 25 ounces Check: 25 ounces 0 gallons = 2 / / / 0 / 6 = 5 ounces 6 gallons 5 5 Eample : A 10 foot tree has a 4 foot shadow. How long a shadow will a 25 foot tree have? The ratio of tree to shadow is 10 feet to 4 feet 10 foot tree 4 foot shadow We have a 25 foot tree with an unknown shadow 10 foot tree 25 foot tree = 4 foot shadow foot shadow 10 = =100 =10 foot shadow Math 100 Section 2 Page Eitel

5 Eample 4 : The scale on a map states that on the map is equal to miles in the real world. If the distance on a map is 10 inches apart how many miles are the actual cities apart? Does it matter which of the two values goes on the top of the ratio? When we say that the ratio of is equal to miles we mean that on a map equals miles on land. We can compare inches to miles or miles to inches and the answer will still be the same as long as we keep the matching units in the numerator and matching units in the denominator. = miles miles = inches y miles can be written or miles = miles y inches but the fraction we set the ratio equal to must have the units in the same place as the first ratio Putting inches over miles The ratio of inches to miles is to miles miles They are 10 inches apart for an unknown number of miles Putting miles over inches The ratio of inches to miles is to miles miles They are 10 inches apart for an unknown number of miles miles = 10 inches miles miles = miles 10 inches 4 = 10 4 = 0 4 = 10 4 = = 0 4 = 7.5 miles 4 4 = 0 4 = 7.5 miles Math 100 Section 2 Page Eitel

6 Eample 5 : A private college advertises that they have a ratio of one teacher for every 18 full-time students. The college plans to enroll 216 students. How many teachers must be employed? The ratio of teachers to students is 1 teacher to 18 students There are 216 students for an unknown number of teachers 1 teacher 18 students = teachers 216 students 18 = = = =12 teachers Check: 12 teachers 216 students = 1 / 2 / 1 2 / 1 / / 6 18 Math 100 Section 2 Page Eitel

GEARING UP EXAMPLES. 4 to 3 4:3

GEARING UP EXAMPLES. 4 to 3 4:3 GEARING UP EXAMPLES B 2 Teeth A 8 Teeth DEFINITION - RATIO As gear A revolves times, it will cause gear B to revolve times. Hence, we say that gear ratio of A to B is to. In mathematics, a ratio is a comparison