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Multi-Step
Rational Number Problems

Master solving real-world problems with fractions, decimals, and integers by combining operations, converting forms, and checking your work with estimation.

7th Grade Math

What Are Rational Numbers?

A rational number is any number that can be written as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\). This includes whole numbers, fractions, mixed numbers, decimals (terminating or repeating), and negative versions of all of these.

Integers
\(-5,\; 0,\; 12\) are all rational: \(\frac{-5}{1},\; \frac{0}{1},\; \frac{12}{1}\)
Fractions
\(\frac{3}{4},\; -\frac{7}{2},\; \frac{11}{5}\) are already in \(\frac{a}{b}\) form
Decimals
\(0.75 = \frac{3}{4}\) and \(0.\overline{3} = \frac{1}{3}\)

Why Multi-Step Problems?

Real life rarely asks you to do just one operation. Buying groceries, splitting a bill, or computing a raise all require chaining multiple operations together: addition, subtraction, multiplication, and division with positive and negative rational numbers.

Read & Identify Convert Forms Compute Step-by-Step Estimate & Verify

Three Forms of the Same Number

Every rational number can be expressed as a fraction, a decimal, or a percent. Converting between forms is a key skill for multi-step problems.

Same Value Fraction
\(\frac{3}{4}\)
 Decimal
\(0.75\)
 Percent
\(75\%\)
 ÷ ÷ ×100

Core Rule: Properties of Operations

When solving multi-step problems, you can use properties of operations to simplify before computing. These properties work for all rational numbers: positive, negative, fractions, and decimals.

1
Commutative: \(a + b = b + a\) and \(a \times b = b \times a\). Rearrange to combine friendlier numbers first.
2
Associative: \((a + b) + c = a + (b + c)\). Regroup to make mental math easier.
3
Distributive: \(a(b + c) = ab + ac\). Break apart difficult multiplications into simpler parts.

Order of Operations Still Applies

Multi-step problems require strict order of operations (PEMDAS/GEMS), even when the numbers are fractions, decimals, or negative. The rules do not change: the numbers just get trickier.

Grouping ( ) Exponents powers Multiply ÷ Add − Left to Right

Converting vs. Computing

Students often confuse when to convert between forms and when to start computing. The rule: convert first so all numbers are in the same form, then compute.

Converting Computing
What Change form (fraction ↔ decimal) Perform operations \((+, -, \times, \div)\)
When Before computing, if forms are mixed After all numbers share a form
Example \(\frac{1}{4} \to 0.25\) \(0.25 + 0.75 = 1.00\)
Goal Make operations easier Get the answer

The 5-Step Method

1
Read carefully. Identify what you are solving for and underline key numbers.
2
Identify the forms. Are numbers given as fractions, decimals, percents, or a mix?
3
Convert as needed. Put all numbers in the same form that makes computation easiest.
4
Compute step-by-step. Follow order of operations. Show each step clearly.
5
Estimate and verify. Use rounding or mental math to check: does the answer make sense?

Example 1: The Percent Raise

A woman earns \(\$25\) per hour and receives a \(10\%\) raise. What is her new hourly wage?

1
Convert percent to a decimal: \(10\% = 0.10\)
2
Compute the raise: \(\$25 \times 0.10 = \$2.50\)
3
Add to the original: \(\$25.00 + \$2.50 = \$27.50\)

Example 2: Centering a Towel Bar

A towel bar is \(9\frac{3}{4}\) inches long. The door is \(27\frac{1}{2}\) inches wide. How far from each edge should the bar be placed to be centered?

1
Subtract bar from door: \(27\frac{1}{2} - 9\frac{3}{4}\)
2
Convert to common denominator: \(27\frac{2}{4} - 9\frac{3}{4} = 26\frac{6}{4} - 9\frac{3}{4} = 17\frac{3}{4}\)
3
Divide remaining space by 2: \(17\frac{3}{4} \div 2 = \frac{71}{4} \times \frac{1}{2} = \frac{71}{8} = 8\frac{7}{8}\) inches

Example 3: Negative Numbers in Real Life

A scuba diver starts at \(-12.5\) feet (below sea level). She descends \(8\frac{3}{4}\) more feet, then ascends \(6.25\) feet. What is her final depth?

0 ft
\(-12.5\text{ ft}\)
 Start
\(-8.75\)
\(-21.25\text{ ft}\)
\(+6.25\)
\(-15.0\text{ ft}\)
 Final

Checking with Estimation

After computing an exact answer, always estimate to verify reasonableness. Round each number to a friendly value, perform the operation mentally, and compare.

Round Fractions
\(\frac{7}{8} \approx 1\), \(\frac{1}{3} \approx 0.33\), \(\frac{3}{4} \approx 0.75\). Use benchmark fractions: \(0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1\).
Round Decimals
\(4.87 \approx 5\), \(-3.14 \approx -3\). Round to the nearest whole number for a quick mental check.
Use Compatible Numbers
\(49.6 \div 7.1 \approx 49 \div 7 = 7\). Pick numbers that divide evenly to estimate quotients.

Handling Mixed Forms in One Problem

When a problem mixes fractions, decimals, and percents, choose one form and convert everything before computing. Which form is best?

1
Choose decimals when all fractions have denominators of 2, 4, 5, 8, 10, 20, 25, or 100: they convert to clean decimals.
2
Choose fractions when denominators are 3, 6, 7, 9, 11, etc.: these create repeating decimals that are hard to work with.
3
Convert percents to decimals (divide by 100) or fractions \(\left(\frac{p}{100}\right)\) based on which form the other numbers use.

Distributive Property Shortcut

The distributive property is especially powerful for mental math with rational numbers. Break apart a multiplication into two easier pieces.

\(6 \times 4.5\)
 distribute
\(6 \times 4 = 24\)
\(6 \times 0.5 = 3\)
\(24 + 3 = 27\)

Common Mistakes to Avoid

✕ Sign Errors
Forgetting that subtracting a negative is the same as adding: \(5 - (-3) = 5 + 3 = 8\), not \(5 - 3 = 2\).
✕ Wrong Order
Adding before multiplying: \(2 + 3 \times 4 = 14\), not \(20\). Always multiply/divide first.
✕ Bad Conversions
Writing \(\frac{1}{3} = 0.3\) instead of \(0.\overline{3}\). Truncating repeating decimals causes errors.
✕ Skipping Estimation
Not checking if the answer is reasonable. An estimate catches errors like misplaced decimals or flipped signs.

Strategy Summary

Step Action Key Tip
1. Read Identify the unknowns and given values Underline key numbers
2. Convert Put all values in the same form Decimals for clean fractions, fractions for repeating
3. Plan Decide which operations and what order Write an expression before computing
4. Compute Apply order of operations carefully Show every step
5. Check Estimate with rounded values If estimate and answer disagree: redo

Question 1 of 5

A shirt costs \(\$40\). It is on sale for \(25\%\) off, and then you use a coupon for an additional \(\$5\) off. What is the final price?

Question 2 of 5

Evaluate: \(-3.5 + 2 \times 1.5 - (-0.75)\)

Question 3 of 5

A recipe calls for \(2\frac{1}{3}\) cups of flour. If you are making \(1\frac{1}{2}\) times the recipe, how much flour do you need?

Question 4 of 5

The temperature was \(-4.8^\circ\text{F}\) in the morning. By noon it rose \(12\frac{1}{4}^\circ\text{F}\), then dropped \(3.6^\circ\text{F}\) by evening. What was the evening temperature?

Question 5 of 5

Which expression uses the distributive property to correctly simplify \(8 \times 3\frac{3}{4}\)?