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Solve non-routine multi-step problems

Non-routine multi-step problem solving involves interpreting a situation, identifying relevant and missing quantities, representing relationships with equations, tables, diagrams, or bar models, and selecting and sequencing operations—including work with whole numbers, fractions, decimals, ratios, rates, and percents—to reach a justified solution. It includes recognizing that different strategies may be valid, checking units and reasonableness, and explaining why the result satisfies all conditions, without requiring advanced algebraic generalization or highly complex numerical cases.

Detailed Explanation: Solve non-routine multi-step problems

When a problem has several conditions, do not start calculating immediately. First:

  1. Identify what the question asks.
  2. List the quantities you know.
  3. Decide which quantity must be found first.
  4. Write the operations in order.
  5. Check that the answer fits every condition.

Example

A class has (20)(20) students and 22 adults going to a museum.

  • A student ticket costs 8$.
  • An adult ticket costs (25%)(25\%) more than a student ticket.
  • The museum gives a (10%)(10\%) discount on the ticket total.
  • The bus costs 238$.
  • The school pays 250$ of the total cost.
  • The students equally share the rest.

How much should each student pay?

Step 1: Find the adult ticket price

The adult ticket is (25%)(25\%) more than 8$.

25% of 8=0.25×8=$225\% \text{ of } 8=0.25 \times 8=\$2

So an adult ticket costs

8+2=$108+2=\$10

Step 2: Find the ticket cost before the discount

Student tickets:

20×8=$16020 \times 8=\$160

Adult tickets:

2×10=$202 \times 10=\$20

Total before the discount:

160+20=$180160+20=\$180

Step 3: Apply the (10%)(10\%) discount

Find the discount:

10% of 180=0.10×180=$1810\% \text{ of } 180=0.10 \times 180=\$18

Subtract the discount from the ticket total:

180−18=$162180-18=\$162

The discounted tickets cost 162$. The discount applies to the tickets, not to the bus.

Step 4: Add the bus cost

162+238=$400162+238=\$400

The complete trip costs 400$.

Step 5: Subtract the school’s payment

The students must share what remains:

400−250=$150400-250=\$150

Step 6: Divide equally among the students

There are (20)(20) students:

150÷20=$7.50150 \div 20=\$7.50

Each student should pay

$7.50\boxed{\$7.50}

Check

The students pay

20×$7.50=$15020 \times \$7.50=\$150

Adding the school’s payment gives

$150+$250=$400\$150+\$250=\$400

This matches the total trip cost, so the answer is reasonable and satisfies all the conditions.

Learn by doing: Solve non-routine multi-step problems

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