Lesson 12: Problems About Comparing, Elapsed-Time Problems

New Concept

Comparison problems find the difference between two values. We solve by subtracting the lesser number from the greater or an earlier time from a later one.

Comparison Formula

Elapsed-Time Formula

What’s next

This lesson provides the foundation for solving word problems. Soon, you'll apply these formulas to worked examples involving numerical comparisons and elapsed time calculations.

Property

The difference is found by subtracting the lesser number from the greater number.

Examples

• How many more employees work at a factory with 1320 people than one with 897? $1320 - 897 = 423$ employees.
• The number 620,000 is how much less than 1,000,000? $1,000,000 - 620,000 = 380,000$.
• How much taller is a 58-inch sibling than a 55-inch sibling? $58 - 55 = 3$ inches.

Explanation

Think like a detective! When a problem asks 'how much greater?' or 'how much less?', you're hunting for the 'difference.' This is just the gap between two numbers. To crack the case, you simply need to subtract the smaller value from the bigger one. It’s the easiest way to solve the mystery!

Property

Elapsed time is the length of time between two points in time. Use the formula:

Examples

• How many years were there from the year 1492 to 1776? $1776 - 1492 = 284$ years.
• Dr. Martin Luther King, Jr. was 34 in 1963. In what year was he born? $1963 - 34 = 1929$.
• John F. Kennedy was born in 1917 and elected president at age 43. In what year was he elected? $1917 + 43 = 1960$.

Explanation

Welcome, time traveler! To find out how much time has passed between two moments, you just need a simple subtraction trick. Think of it like a comparison problem, but with dates. Just subtract the earlier date from the later one to find the difference. It's how you calculate your own age from your birthdate!

Property

When adding a long sequence of numbers, you can find pairs of addends that have the same sum and multiply that sum by the number of pairs to find the total quickly.

Examples

• Sum the first ten even numbers (2 to 20): Pair $2+20=22$, $4+18=22$, etc. There are 5 pairs, so $5 \times 22 = 110$.
• Find the sum of all integers from 1 to 10: Pair $1+10=11$, $2+9=11$, etc. There are 5 pairs, so $5 \times 11 = 55$.
• Sum the numbers 5, 10, 15, 20, 25, 30: Pair $5+30=35$, $10+25=35$, $15+20=35$. There are 3 pairs, so $3 \times 35 = 105$.

Explanation

Don't get stuck adding long lists of numbers one by one! Be a math ninja and find a pattern. By pairing the first and last numbers, then the second and second-to-last, and so on, you can often create groups with the same sum. This turns a long, boring addition problem into easy multiplication!