Multi-Step Word Problems with Rational Numbers

Real-world math often requires more than one step and mixes different types of numbers, such as whole numbers, fractions, and decimals. To solve these multi-step word problems, you need to carefully plan your operations and be comfortable converting between number formats.

Steps to Solve Multi-Step Problems

1. Read and Understand: Identify what the problem is asking you to find.
2. Gather Information: Pull out the key numbers (fractions, decimals, or whole numbers).
3. Plan the Math: Decide which operations (addition, subtraction, multiplication, or division) you need and in what order.
4. Convert if Necessary: If a problem mixes decimals and fractions, it is often easier to convert them all to one format before calculating.
5. Solve and Check: Perform the calculations step-by-step and verify if your final answer makes logical sense.

Example 1: Working with Fractions

Problem: A tank is $\frac{3}{4}$ full with $450$ liters of water. How many liters does a full tank hold?

Step 1: Understand the relationship. The problem tells us that $\frac{3}{4}$ of the total capacity equals $450$ liters.

Step 2: Set up the equation. Let $T$ be the total capacity. $\frac{3}{4} \times T = 450$

Step 3: Solve for $T$. To find the total, divide the current volume by the fraction: $T = 450 \div \frac{3}{4}$

When dividing by a fraction, multiply by its reciprocal: $T = 450 \times \frac{4}{3} = \frac{450 \times 4}{3} = \frac{1800}{3} = 600$

Answer: A full tank holds $600$ liters.

Example 2: Working with Decimals

Problem: Tom walks $2.5$ km at a fare of \0.35$per km. He pays with a$$5$ bill. What is his change?

Step 1: Calculate the total fare. Multiply the distance by the cost per kilometer: $2.5 \times 0.35 = 0.875$ Tom's total fare is \0.875$.

Step 2: Calculate the change. Subtract the total fare from the amount Tom paid (\5$). Remember to line up the decimals by writing$5$as$5.000$: $5.000 - 0.875 = 4.125$

Answer: Tom's change is \4.125$(or$$4.13$ if rounded to the nearest cent).