Lesson 15: Problems About Equal Groups

New Concept

Problems about equal groups involve multiplication. We find a total by multiplying the number of groups by the number of items in each group.

Number of groups × number in group = total

What’s next

Next, you'll apply this foundational pattern to solve for unknown totals, group sizes, or the number of groups in various word problems.

Property

Number of groups × number in group = total, which can be written as the formula:

Examples

$15 \text{ rows} \times 20 \text{ chairs per row} = 300 \text{ total chairs}$
$450 \text{ cents} \div 25 \text{ cents per cup} = 18 \text{ cups sold}$
$12 \text{ rows} \times 18 \text{ parking spaces per row} = 216 \text{ total spaces}$

Explanation

Think of this as a super-fast way to handle lots of identical sets! Instead of adding the same number over and over, you just multiply. If you have 10 rows of chairs with 20 chairs in each, multiplying is way quicker than counting every single one. The phrase 'in each' is your secret code word for these problems!

Property

In an equal groups problem, if the total is known but a factor (the number of groups or the number in each group) is unknown, we use division to find it. If $n \times g = t$, then $n = t \div g$.

Examples

$\text{Problem: } 232 \text{ students in } 8 \text{ classrooms.} \rightarrow 232 \div 8 = 29 \text{ students per classroom.}$
$\text{Problem: } 1200 \div w = 300 \rightarrow w = 1200 \div 300 = 4$
$\text{Problem: } 63w = 63 \rightarrow w = 63 \div 63 = 1$

Explanation

Imagine you're a detective cracking a code! You have the final answer (the total) and one clue (the number of groups or items in each). To find the missing piece, you use division—the ultimate reverse-multiplication tool. This lets you work backward to uncover the unknown number and declare 'case closed' on any 'equal groups' mystery you face.