4.3 Solving Proportions and Applications of Proportions 1 Solving equations of the form a  x = b Before we begin solving proportions, we’ll begin by solving some simpler equations. Notation: a and b are numbers and x is unknown. 1. Our goal is to get x by itself. Five times what number is 40? Answer: 8. Twelve times what number is 36? Answer: 3. The answer will not always be obvious, so we will need a procedure. 2. Draw a line under each side to indicate division. 3. Write the value of ‘a’ below each line and simplify each side. 3 3 x = 19 Your Turn Problem #1

4.3 Solving Proportions and Applications of Proportions 2 1. Draw a line under each side to indicate division. 2. Write the value of ‘a’ below each line and simplify each side. If it doesn’t divide evenly, simplify the fraction on the right hand side. 18 Your Turn Problem #2

4.3 Solving Proportions and Applications of Proportions 3 Solving equations of the form a  x = b where a is a proper fraction In the last two examples, ‘a’ was a whole number. To solve, we divided by ‘a’ on both sides. In the next problem, ‘a’ will be a fraction. The procedure is still the same, divide by ‘a’ on both sides. However, when we divide by ‘a’ on both sides, this is equivalent to multiplying by the reciprocal of ‘a’ on both sides. 1. Rewrite the problem and multiply by the reciprocal of ‘a’ on both sides. 2. The numbers on the left hand side (LHS) all divide out. Simply and multiply on the RHS. Your Turn Problem #3

4.3 Solving Proportions and Applications of Proportions 4 1. Rewrite the problem and multiply by the reciprocal of ‘a’ on both sides. 2. The numbers on the left hand side (LHS) all divide out. Simply and multiply on the RHS. Your Turn Problem #4

4.3 Solving Proportions and Applications of Proportions 5 Your Turn Problem #5 Solving equations of the form a  x = b where ‘a’ and or b are decimals 1. Draw a line under each side to indicate division. 2. Write the value of ‘a’ below each line and divide..04

4.3 Solving Proportions and Applications of Proportions 6 Solving a Proportion When one of the terms of a proportion is unknown, we can “solve” the proportion for that term. The solution is the value of the unknown that makes the proportion true. Procedure: Solving a Proportion Step 1. Multiply the cross product which has the variable. Step 2. Write an equal sign after the cross product. Step 3. Multiply the other cross product. Write it after the equal sign. Step 4. Solve the equation. 1. Multiply the cross product with the variable. 2. Write an equal sign after the cross product. Your Turn Problem #6 3. Multiply the other cross product. 4. Solve the equation.

4.3 Solving Proportions and Applications of Proportions 7 1. Multiply the cross product with the variable. 2. Write an equal sign after the cross product. Your Turn Problem #7 3. Multiply the other cross product. 4. Solve the equation.

4.3 Solving Proportions and Applications of Proportions 8 Applications of Proportions Proportions are a useful tool for solving problems in various disciplines such as business, science, engineering, medicine, and other sciences. The following steps may be used to solve an application problem using a proportion. Keep in mind, that like units must be in their respective numerators and denominators of the ratios when setting up the proportion. Steps for Solving an Application Problem Using a Proportion Step 1. Read and understand the problem. Assign a letter to represent the unknown quantity. Step 2.Set up a proportion. Keep like units in their respective numerators and denominators. (i.e the units match across) Step 3. Solve the proportion. Step 4. Answer the question. Example 8. A patient is to receive 8 cc of medication every 5 hours, at this rate, how many cc of medication should be administered in 90 hours? Step 1. Let x amount of cc to be administered Step 2. Set up proportion (match units across) Step 3. Solve the proportion. Step 4. Answer the question. The patient will receive 144 cc of the medication. Your Turn Problem #8 Em traveled 180 miles on 12 gallons of gas. At this rate, how many gallons of gas would be required to travel 330 miles? Answer: 22 gallons The End B.R