Learn on PengienVision, Mathematics, Grade 7Chapter 1: Integers and Rational Numbers

Lesson 10: Solve Problems with Rational Numbers

In this Grade 7 lesson from enVision Mathematics Chapter 1, students learn how to solve multi-step problems involving rational numbers by selecting appropriate operations such as addition, subtraction, multiplication, and division. The lesson applies the Distributive Property and other properties of operations to real-world contexts including temperature change, elevation, and financial calculations. Students practice deciding which rational number operations to use based on problem structure and reasoning.

Section 1

Solving Real-World Problems with Rational Numbers

Property

Solve real-world and mathematical problems involving the four operations with rational numbers.
This requires translating a real-world scenario into a mathematical expression using addition, subtraction, multiplication, or division of rational numbers.

Examples

A baker has $4\frac{1}{2}$ pounds of flour. A cake recipe requires $2\frac{3}{4}$ pounds. How much flour is left? $\frac{9}{2} - \frac{11}{4} = \frac{18}{4} - \frac{11}{4} = \frac{7}{4}$ , or $1\frac{3}{4}$ pounds.
A submarine at the surface dives $20\frac{1}{2}$ meters, then rises $8\frac{1}{4}$ meters. What is its new depth? $-20\frac{1}{2} + 8\frac{1}{4} = -\frac{41}{2} + \frac{33}{4} = -\frac{82}{4} + \frac{33}{4} = -\frac{49}{4}$ , or $-12\frac{1}{4}$ meters.
Three friends share a pizza. Anna eats $\frac{1}{4}$ , and Ben eats $\frac{1}{3}$ . What fraction of the pizza did they eat combined? $\frac{1}{4} + \frac{1}{3} = \frac{3}{12} + \frac{4}{12} = \frac{7}{12}$ of the pizza.

Explanation

Rational numbers are used to handle everyday tasks like measuring ingredients, tracking distances, or splitting bills. The first step is to read the problem carefully and decide which of the four basic operations is needed to solve it.

Section 2

Multi-step Problems with Rational Numbers

Property

Solve multi-step real-life problems with rational numbers in any form (whole numbers, fractions, and decimals) by breaking them down into a sequence of smaller calculations. Work forwards, using the result of one step as the input for the next step.

Examples

A 200 dollars tablet is on sale for 10% off. Then, an 8% sales tax is applied to the sale price. The sale price is $200(0.90) = 180$ dollars. The final cost with tax is $180(1.08) = 194.40$ dollars.

A person earns 60,000 dollars and pays 20% in taxes. Of the remaining amount, they spend $\frac{1}{4}$ on rent. After-tax income is $60000(0.80) = 48000$ dollars. Rent is $\frac{1}{4} \times 48000 = 12000$ dollars.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Solving Real-World Problems with Rational Numbers

Property

Examples

A baker has $4\frac{1}{2}$ pounds of flour. A cake recipe requires $2\frac{3}{4}$ pounds. How much flour is left? $\frac{9}{2} - \frac{11}{4} = \frac{18}{4} - \frac{11}{4} = \frac{7}{4}$ , or $1\frac{3}{4}$ pounds.
A submarine at the surface dives $20\frac{1}{2}$ meters, then rises $8\frac{1}{4}$ meters. What is its new depth? $-20\frac{1}{2} + 8\frac{1}{4} = -\frac{41}{2} + \frac{33}{4} = -\frac{82}{4} + \frac{33}{4} = -\frac{49}{4}$ , or $-12\frac{1}{4}$ meters.
Three friends share a pizza. Anna eats $\frac{1}{4}$ , and Ben eats $\frac{1}{3}$ . What fraction of the pizza did they eat combined? $\frac{1}{4} + \frac{1}{3} = \frac{3}{12} + \frac{4}{12} = \frac{7}{12}$ of the pizza.

Explanation

Section 2

Multi-step Problems with Rational Numbers

Property

Examples

A 200 dollars tablet is on sale for 10% off. Then, an 8% sales tax is applied to the sale price. The sale price is $200(0.90) = 180$ dollars. The final cost with tax is $180(1.08) = 194.40$ dollars.

A person earns 60,000 dollars and pays 20% in taxes. Of the remaining amount, they spend $\frac{1}{4}$ on rent. After-tax income is $60000(0.80) = 48000$ dollars. Rent is $\frac{1}{4} \times 48000 = 12000$ dollars.

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Solving Real-World Problems with Rational Numbers

Property

Examples

A baker has $4\frac{1}{2}$ pounds of flour. A cake recipe requires $2\frac{3}{4}$ pounds. How much flour is left? $\frac{9}{2} - \frac{11}{4} = \frac{18}{4} - \frac{11}{4} = \frac{7}{4}$ , or $1\frac{3}{4}$ pounds.
A submarine at the surface dives $20\frac{1}{2}$ meters, then rises $8\frac{1}{4}$ meters. What is its new depth? $-20\frac{1}{2} + 8\frac{1}{4} = -\frac{41}{2} + \frac{33}{4} = -\frac{82}{4} + \frac{33}{4} = -\frac{49}{4}$ , or $-12\frac{1}{4}$ meters.
Three friends share a pizza. Anna eats $\frac{1}{4}$ , and Ben eats $\frac{1}{3}$ . What fraction of the pizza did they eat combined? $\frac{1}{4} + \frac{1}{3} = \frac{3}{12} + \frac{4}{12} = \frac{7}{12}$ of the pizza.

Explanation

Section 2

Multi-step Problems with Rational Numbers

Property

Examples

A 200 dollars tablet is on sale for 10% off. Then, an 8% sales tax is applied to the sale price. The sale price is $200(0.90) = 180$ dollars. The final cost with tax is $180(1.08) = 194.40$ dollars.

A person earns 60,000 dollars and pays 20% in taxes. Of the remaining amount, they spend $\frac{1}{4}$ on rent. After-tax income is $60000(0.80) = 48000$ dollars. Rent is $\frac{1}{4} \times 48000 = 12000$ dollars.