Learn on PengiThe Art of Problem Solving: Prealgebra (AMC 8)Chapter 5: Equations and Inequalities

Lesson 4: Word Problems

In this Grade 4 lesson from The Art of Problem Solving: Prealgebra (AMC 8), students learn a structured method for translating word problems into algebraic equations by assigning variables to unknown quantities and expressing multiple unknowns in terms of a single variable. Lessons cover setting up and solving linear equations from verbal descriptions, including problems with fractions, sums, and real-world scenarios. Students also practice checking solutions to confirm their equations correctly represent the original problem.

Section 1

Defining Variables and Equations

Property

It is very important to specify precisely what the variable represents. The variable must stand for a number. For example, use $w$ for "Lima's weight," not for "Lima."
Although the equation includes the variable, the two sides of the equation may actually be expressions for some other quantity, such as a ratio.

Examples

To describe "Sam is 5 years older than Tim," define $S$ as "Sam's age in years" and $T$ as "Tim's age in years." The correct equation is $S = T + 5$ . Defining variables just as "Sam" and "Tim" would be unclear.

If a school has a student-to-teacher ratio of 15 to 1 and there are 450 students, we want to find the number of teachers, $t$ . The equation $\frac{450}{t} = \frac{15}{1}$ is about the ratio, not the number of teachers itself.

Section 2

Problems with Two Unknowns

Property

When a problem involves two unknown quantities, name the first unknown with a variable (e.g., $n$ ).
Then, read the problem to find the relationship between the two unknowns and write an expression for the second unknown using the same variable (e.g., $n+5$ or $2n-10$ ).
Finally, form an equation that relates both unknowns and solve.

Examples

One number is ten more than another. Their sum is 52. Find the numbers. Let the first number be $n$ . The second is $n+10$ . The equation is $n + (n+10) = 52$ . This simplifies to $2n=42$ , so $n=21$ . The numbers are 21 and 31.

The sum of two numbers is negative twenty. One number is six less than the other. Find the numbers. Let the first number be $x$ . The second is $x-6$ . The equation is $x + (x-6) = -20$ . This simplifies to $2x = -14$ , so $x=-7$ . The numbers are -7 and -13.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Defining Variables and Equations

Property

It is very important to specify precisely what the variable represents. The variable must stand for a number. For example, use $w$ for "Lima's weight," not for "Lima."
Although the equation includes the variable, the two sides of the equation may actually be expressions for some other quantity, such as a ratio.

Examples

To describe "Sam is 5 years older than Tim," define $S$ as "Sam's age in years" and $T$ as "Tim's age in years." The correct equation is $S = T + 5$ . Defining variables just as "Sam" and "Tim" would be unclear.

If a school has a student-to-teacher ratio of 15 to 1 and there are 450 students, we want to find the number of teachers, $t$ . The equation $\frac{450}{t} = \frac{15}{1}$ is about the ratio, not the number of teachers itself.

Section 2

Problems with Two Unknowns

Property

Examples

One number is ten more than another. Their sum is 52. Find the numbers. Let the first number be $n$ . The second is $n+10$ . The equation is $n + (n+10) = 52$ . This simplifies to $2n=42$ , so $n=21$ . The numbers are 21 and 31.

The sum of two numbers is negative twenty. One number is six less than the other. Find the numbers. Let the first number be $x$ . The second is $x-6$ . The equation is $x + (x-6) = -20$ . This simplifies to $2x = -14$ , so $x=-7$ . The numbers are -7 and -13.

Overview

Lesson overview

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

Overview

Section 1

Defining Variables and Equations

Property

It is very important to specify precisely what the variable represents. The variable must stand for a number. For example, use $w$ for "Lima's weight," not for "Lima."
Although the equation includes the variable, the two sides of the equation may actually be expressions for some other quantity, such as a ratio.

Examples

To describe "Sam is 5 years older than Tim," define $S$ as "Sam's age in years" and $T$ as "Tim's age in years." The correct equation is $S = T + 5$ . Defining variables just as "Sam" and "Tim" would be unclear.

If a school has a student-to-teacher ratio of 15 to 1 and there are 450 students, we want to find the number of teachers, $t$ . The equation $\frac{450}{t} = \frac{15}{1}$ is about the ratio, not the number of teachers itself.

Section 2

Problems with Two Unknowns

Property

Examples

One number is ten more than another. Their sum is 52. Find the numbers. Let the first number be $n$ . The second is $n+10$ . The equation is $n + (n+10) = 52$ . This simplifies to $2n=42$ , so $n=21$ . The numbers are 21 and 31.

The sum of two numbers is negative twenty. One number is six less than the other. Find the numbers. Let the first number be $x$ . The second is $x-6$ . The equation is $x + (x-6) = -20$ . This simplifies to $2x = -14$ , so $x=-7$ . The numbers are -7 and -13.