Math Core

Real problems rarely take just one step. You might multiply to find a total, subtract what was used, and then divide. The trick is to slow down, plan the steps, and check that your answer makes sense.

A plan for multi-step problems

Four steps

Understand. What do you know? What does the question ask?
Plan. Which operations, in what order? Write an equation with a letter for the unknown.
Solve. Do one step at a time.
Check. Estimate to see if the answer is reasonable. If you divided, decide what the remainder means.

A letter such as $n$ can stand for the number you are looking for. Writing an equation first helps you see all the steps at once. Parentheses show which step comes first.

Worked examples

Worked example: Multiply, then divide

A teacher buys 3 packs of pencils with 24 pencils in each pack. She shares them equally among 8 tables. How many pencils does each table get?

Plan. First find the total, then share it. Let $p$ be the number of pencils per table.

p = (3 \times 24) \div 8

Solve.

$3 \times 24 = 72$ pencils in all.
$72 \div 8 = 9$ .

Each table gets 9 pencils.

Check. $9 \times 8 = 72$ and $72 = 3 \times 24$ . ✓

Worked example: Three steps and a remainder

A farmer picks 245 apples. He sells 5 bags with 30 apples in each bag. He packs the rest into boxes of 4. How many boxes can he fill completely?

Plan. Find how many apples were sold, subtract, then divide. Let $b$ be the number of full boxes.

b = (245 - 5 \times 30) \div 4

Solve.

Sold: $5 \times 30 = 150$ apples.
Left: $245 - 150 = 95$ apples.
Boxes: $95 \div 4 = 23 \text{ R } 3$ , because $4 \times 23 = 92$ .

The question asks for full boxes, so drop the remainder. He fills 23 boxes, with 3 apples left over.

Worked example: Estimate to check

A class trip costs $1,350 in all. The bus costs $270. The rest is split equally among 9 parent helpers. How much does each helper pay?

Plan. Let $h$ be the amount each helper pays.

h = (1{,}350 - 270) \div 9

Solve.

$1{,}350 - 270 = 1{,}080$ .
$1{,}080 \div 9$ : 10 hundreds $\div$ 9 = 1 hundred, with 1 hundred left. That's 18 tens, and 18 tens $\div$ 9 = 2 tens. 0 ones $\div$ 9 = 0. So $1{,}080 \div 9 = 120$ .

Each helper pays $120.

Check. Round: about $1,400 minus $300 is $1,100. Split among about 10 people, that's about $110 each. $120 is close, so the answer is reasonable.

Common mistake

Answer the question that was asked. In the apple problem, 95 is the number of apples left, not the answer. Before you stop, reread the last sentence of the problem and make sure your number matches it.

Tip

Label each step with words ("pencils in all", "apples left"). It keeps you from dividing or subtracting the wrong number.

Practice

Practice 1

A class has 4 boxes of crayons with 36 crayons in each box. The crayons are shared equally among 6 students. How many crayons does each student get?

Type a number.

Practice 2

Nora has 120 stickers. She gives 24 to her brother. She puts the rest on 8 pages of her album, with the same number on each page. How many stickers go on each page?

Type a number.

Practice 3

Sam has 90 trading cards. He buys 3 packs with 10 cards in each pack. Then he shares all his cards equally among 4 friends. Which equation finds $c$ , the number of cards each friend gets?

Practice 4

At a concert, 315 students and 24 teachers need seats. Each row has 8 seats. How many rows are needed so everyone can sit?

Type a number.

Practice 5

A baker makes 6 trays of cookies with 45 cookies on each tray. She sells 185 cookies. She puts the rest in bags of 8. How many bags can she fill completely?

Type a number.

Practice 6

Maria wants a bike that costs $150. She has already saved $38. She saves $8 more each week. How many more weeks until she has exactly enough?

Type a number.

Practice 7

Liam read 48 pages. Ana read 4 times as many pages as Liam. She read the same number of pages each day for 6 days. How many pages did Ana read each day?

Type a number.