Solving Word Problems with Proportions: Step-by-Step

Welcome to understanding proportions! Today we'll explore how equal ratios appear in everyday life.A proportion is simply an equation that shows two ratios are equal to each other.Let's look at a common example: doubling a recipe. If we need twice as much food, every ingredient must be doubled to maintain the same proportions.Maps use proportions to show distances. A scale of one inch equals one hundred miles means the same ratio applies anywhere on the map.Scale models, like model cars, use proportions to maintain the correct shape. Every measurement is scaled by the same ratio.There are several key phrases that signal when a problem involves proportions.Look for words like 'per', 'scale', 'ratio', 'rate', and phrases like 'times as many'.Proportions often involve quantities that change together, like time and distance when driving at a constant speed.Now that we understand what proportions are and where to find them, let's learn how to set them up and solve them.Let's examine this word problem about paint coverage.First, we identify the important numbers and their relationships. We have 3 gallons covering 750 square feet, and need to find how many gallons for 2000 square feet.When setting up a proportion, we need to organize corresponding parts correctly. Let's create our proportion.Notice how we align the parts: gallons on top, square feet on bottom. This keeps our units consistent.Corresponding parts must match. The numerators both represent gallons of paint, while the denominators both represent square feet of coverage.Let's review the key rules for setting up proportions correctly.When placing our unknown variable x, it must go in the position that corresponds to what we're solving for - in this case, the number of gallons needed.Our final proportion is now properly set up and ready to solve: three over seven hundred fifty equals x over two thousand.To solve a proportion using cross multiplication, we multiply the numerator of one fraction by the denominator of the other.We multiply 3 times 24 on one side, and 8 times x on the other side.This gives us the equation 72 equals 8x. To solve for x, we divide both sides by 8.And we get x equals 9.Let's try an example with units. Here we have kilometers per hour.When we cross multiply, the units will cancel out because they appear in both numerator and denominator.Let's solve step by step. First, multiply 15 times 5 equals 2x.This gives us 75 equals 2x.Dividing both sides by 2.We get x equals 37.5 kilometers.Now let's solve a proportion with decimals.We cross multiply 2.5 times 12 equals 4 times x.This gives us 30 equals 4x.Dividing both sides by 4.Our final answer is x equals 7.5.Let's verify our solution to this proportion problem.We solved this and found that x equals 8 gallons. Now let's verify if this answer is correct.There are three key steps to verify our answer.First, let's plug our answer back into the original proportion.Next, we check that our units match on both sides of the equation.Finally, we apply common sense to check if our answer is reasonable.Remember, even if the math checks out, your answer must make sense in the real world.By following these verification steps, we can be confident in our solution.Let's solve our first proportion problem about scaling a recipe.We need to find how much flour is needed for ten people when the original recipe uses two cups for four people.Using cross multiplication, we multiply two times ten equals four x. Solving for x gives us five cups of flour.Our next example involves calculating travel time based on a constant speed.If we travel one hundred fifty miles in three hours, we can set up a proportion to find the time needed for two hundred fifty miles.Cross multiply and solve: one hundred fifty x equals three times two hundred fifty. The trip will take five hours.Our final example deals with blueprint scaling.If two inches represents ten feet, we can find how many feet five inches represents using the same proportion method.Cross multiply: two x equals ten times five. Solving for x shows that five inches represents twenty-five feet.