Solving Word Problems Using Proportional Equations
Solving Word Problems Using Proportional Equations is a Grade 7 math skill in Reveal Math Accelerated, Unit 2: Proportional Relationships, where students set up and solve proportions in the form a/b = c/d to find unknown quantities in real-world contexts such as scaling recipes, converting units, and solving rate problems. Proportional equations are a cornerstone of 7th grade math and pre-algebra.
Key Concepts
Property A proportion can be used to model and solve real world problems by setting two equivalent ratios equal to each other: $$\frac{a}{b} = \frac{c}{d}$$ where one of the values is an unknown variable.
Examples If 3 apples cost 2 dollars, how much do 9 apples cost? Set up the proportion $\frac{3}{2} = \frac{9}{x}$. Since $3 \times 3 = 9$, multiply the denominator by 3 to find $x = 2 \times 3 = 6$ dollars. A car travels 150 miles on 5 gallons of gas. How many gallons are needed to travel 300 miles? Set up the proportion $\frac{150}{5} = \frac{300}{x}$. Since $150 \times 2 = 300$, multiply the denominator by 2 to find $x = 5 \times 2 = 10$ gallons. A boat travels $80$ miles on $\frac{4}{3}$ gallons of fuel. How many gallons are needed to travel $160$ miles? Set up the proportion $\frac{80}{\frac{4}{3}} = \frac{160}{x}$. Since $80 \times 2 = 160$, multiply the denominator by $2$ to find $x = \frac{4}{3} \times 2 = \frac{8}{3}$ gallons.
Explanation To solve a real world problem using proportions, start by identifying the two quantities being compared. Set up an equation where two ratios are equal, making sure the units align perfectly on both sides of the equal sign. Finally, use ratio reasoning, such as finding a scale factor or applying multiplication, to solve for the unknown value.
Common Questions
How do you solve a word problem using a proportion?
Identify the two ratios that are equal, set them up as a/b = c/d, cross-multiply to get ad = bc, and solve for the unknown variable.
When can you use a proportional equation?
Use a proportion when two quantities are in a constant ratio — such as distance and time at a constant speed, items and total price at a fixed unit rate, or lengths in a scale drawing.
What does it mean to cross-multiply?
Cross-multiplying means multiplying the numerator of each fraction by the denominator of the other. In a/b = c/d, cross-multiplying gives a x d = b x c, which you can then solve for the unknown.
What is Reveal Math Accelerated Unit 2 about?
Unit 2 covers Proportional Relationships, including unit rates, constants of proportionality, tables, graphs, equations, and word problems involving proportional reasoning.