Solving With Proportions
Grade 8 math lesson on solving problems using proportions and cross-multiplication. Students learn to set up proportional equations from word problems and solve for unknown values using the property that cross products are equal in a true proportion.
Key Concepts
Property An alternate method for solving direct variation problems is to write a proportion by setting the two ratios equal to each other. $$ \frac{a 1}{b 1} = \frac{a 2}{b 2} $$.
Examples If 60 people cost 780 dollars, how much for 100? Set up the proportion: $\frac{780}{60} = \frac{x}{100}$. Cross multiply to find $x = 1300$ dollars. If Maggie drives 75 miles in 1.5 hours, how long for 100 miles? $\frac{75}{1.5} = \frac{100}{t}$. Cross multiply to find $t = 2$ hours. If Tom types 100 words in 2.5 minutes, how many in 15? $\frac{100}{2.5} = \frac{w}{15}$. Cross multiply to find $w = 600$ words.
Explanation This is the ultimate shortcut for direct variation problems. You don't even need to find the constant $k$! Just set up two fractions that you know are equal: one with the information you have, and one with the missing piece you want to find. Then, use the magic of cross multiplication to solve for the unknown value.
Common Questions
What is a proportion in math?
A proportion is an equation stating that two ratios are equal. Written as a/b = c/d, a proportion means that the ratio of a to b is the same as the ratio of c to d.
How do you solve a proportion?
To solve a proportion with an unknown, use cross multiplication: multiply the numerator of each fraction by the denominator of the other. Then solve the resulting equation. If a/b = c/d, then a times d = b times c.
When do you use proportions to solve real-world problems?
Proportions are useful when two quantities have a constant ratio. Examples include finding unit prices, scaling recipes, converting units, calculating map distances, and working with similar figures.
What is cross multiplication and how does it work?
Cross multiplication is a shortcut for solving proportions. If a/b = c/d, cross multiply to get a times d = b times c. Then divide both sides by the coefficient of the unknown to find its value.