Lesson 7.10

Word Problems: Sum & Difference

Finally, we answer "When will I use this?" Relationships often come in pairs: "We have 20 people total, but 4 more boys than girls." That's a system!

Introduction

The hardest part of word problems isn't the math—it's the translation. We need to turn English sentences into and equations.

Past Knowledge

Lesson 1.1 (Variables & Phrases). You must recall that "Sum" means add and "Difference" means subtract.

Today's Goal

Write two simple equations from a story and solve them using Elimination.

Future Success

This is the most common type of system found on standardized tests (SAT/ACT).

Key Concepts

Translation Guide

"The Sum is..."

Keywords: Total, Combined, Together, In All.

"The Difference is..."

Keywords: More than, Less than, Difference between.

Note: Usually define as the larger number.

The Elimination Hack

Sum/Difference problems are PERFECT for Elimination. Just add the equations and the usually disappears immediately!

Worked Examples

Example 1: The Number Game

Basic

The sum of two numbers is 20. Their difference is 4. Find the numbers.

Step 1: Translate

Let = large, = small.

Sum:

Diff:

Step 2: Add

Plug in:

Answer: 12 and 8

Example 2: Analyzing Ages

Intermediate

Tom is 5 years older than Jerry. Together, their ages add up to 21. How old is Jerry?

Step 1: Translate

" is older than ": (Wait, this is Substitution form!)

Sum:

Step 2: Substitute

Plug into T.

Jerry is 8.

Example 3: Perimeter Problem

Advanced

The perimeter of a rectangle is 30cm. The length is twice the width. Find dimensions.

Step 1: Formulas

Perimeter:

Relationship:

Step 2: Solve

Plug into L.

→

Width: 5, Length: 10

Common Pitfalls

Subtraction Order

"The difference between x and y" usually means . But "x less than y" means . Be careful with order!

Forgetting the 2

In perimeter problems, students often write . NO! That's the sum of sides. Perimeter adds ALL 4 sides ().

Real-Life Applications

Surveying:

Land surveyors define property lines using these systems.
"The lot length is 3 times the width, and the total boundary is 400ft."