2.2 The Four Operations with Rational Numbers & Multi-Step Real-World Arithmetic Problems
Key Takeaways
- Q.2.a (add, subtract, multiply, divide rational numbers) and Q.2.e (multi-step real-world arithmetic problems) are the highest-volume computational skills tested across the whole GED Math exam.
- Fraction rules: add/subtract need a common denominator; multiply straight across; divide by 'keep-change-flip' (multiply by the reciprocal).
- Integer sign rules: same signs multiply/divide to a positive result, different signs multiply/divide to a negative result — but adding/subtracting integers depends on magnitude, not just sign.
- Multi-step real-world problems must be solved in the correct order — a discount applied before tax gives a different total than tax applied before a discount.
- The first 5 no-calculator questions on the GED test draw heavily from Q.2.a — you must be able to add, subtract, multiply, and divide fractions and integers by hand, quickly and accurately.
Why This Topic Matters
Assessment target Q.2 — "Add, subtract, multiply, divide, and use exponents and roots of rational, fraction and decimal numbers" — has five graded indicators (Q.2.a through Q.2.e), more than any other single target in the Quantitative domain. Two of those indicators, Q.2.a (the four basic operations) and Q.2.e (multi-step real-world arithmetic problems), are the computational backbone of the entire test: virtually every geometry, ratio, statistics, and even algebra item eventually requires you to add, subtract, multiply, or divide a fraction, decimal, or integer correctly. Fumble the arithmetic and a perfectly set-up algebra problem still produces a wrong answer.
This is also the domain most directly tested in the no-calculator section (the first 5 items), so speed and accuracy with fractions and integer signs — done entirely by hand — pay off immediately.
Integer Sign Rules
Multiplication and division follow a clean same-sign/different-sign pattern; addition and subtraction do not — they depend on magnitude.
| Operation | Rule | Example |
|---|---|---|
| Multiply/divide, same signs | Result is positive | (−6) × (−4) = 24 |
| Multiply/divide, different signs | Result is negative | (−6) × 4 = −24 |
| Add, same signs | Add magnitudes, keep the sign | −6 + (−4) = −10 |
| Add, different signs | Subtract magnitudes, keep the sign of the larger magnitude | −6 + 4 = −2 |
| Subtract a negative | Rewrite as adding the opposite | 5 − (−3) = 5 + 3 = 8 |
The most common error on the no-calculator section is dropping or flipping a negative sign midway through a multi-step calculation — always track signs as a separate step before finishing the arithmetic.
Fraction and Decimal Operations
| Operation | Fractions | Decimals |
|---|---|---|
| Add / Subtract | Convert to a common denominator first, then add/subtract numerators | Line up decimal points, then add/subtract as with whole numbers |
| Multiply | Multiply numerators together and denominators together, then simplify | Multiply as whole numbers, then place the decimal point (count total decimal places in both factors) |
| Divide | Keep-Change-Flip: keep the first fraction, change ÷ to ×, flip (find the reciprocal of) the second fraction | Move the decimal point in the divisor to make it a whole number, moving the decimal the same number of places in the dividend, then divide |
Worked example (fraction division): 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8
Mixed numbers must always be converted to improper fractions before multiplying or dividing (though you can add/subtract mixed numbers by working with whole-number and fractional parts separately if the fractional parts don't require borrowing). Example: 2 1/3 as an improper fraction is (2 × 3 + 1)/3 = 7/3.
Worked example (decimal multiplication): 3.2 × 1.5. Multiply as whole numbers: 32 × 15 = 480. The two factors have a total of 2 decimal places (one each), so place the decimal point 2 digits from the right: 4.80, or 4.8.
Worked example (decimal division): 6.4 ÷ 0.08. Move the decimal point in the divisor (0.08) two places right to make it a whole number (8), and move the decimal point in the dividend the same two places (640). Then divide: 640 ÷ 8 = 80.
Mixed-number subtraction with borrowing is a frequent trap: 5 1/4 − 2 3/4. Since 1/4 is smaller than 3/4, you cannot subtract the fractional parts directly. Borrow 1 (as 4/4) from the whole-number part: 5 1/4 becomes 4 5/4. Now subtract: 4 5/4 − 2 3/4 = 2 2/4 = 2 1/2. Students who skip the borrowing step and simply subtract 1/4 − 3/4 as if it were positive produce a wrong, unsimplified answer.
Order of Operations (PEMDAS) in Multi-Step Problems
When an expression mixes operations, follow PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). Q.2.e problems on the GED are almost always real-world word problems requiring multiple steps in a specific order — the test rewards translating the situation into an equation correctly, not just computing quickly.
Worked example: "A phone plan costs $45 per month plus $0.10 per text message over the included 500 texts. If Maria sends 640 texts in a month, what is her total bill?"
- Extra texts: 640 − 500 = 140
- Extra charge: 140 × $0.10 = $14.00
- Total bill: $45 + $14.00 = $59.00
Order-sensitive trap example: "A jacket costs $80. It is marked down 25%, and then a 6% sales tax is applied to the discounted price. What is the final price?"
- Discount amount: $80 × 0.25 = $20, so discounted price = $80 − $20 = $60
- Tax on the discounted price (not the original $80): $60 × 0.06 = $3.60
- Final price: $60 + $3.60 = $63.60
Applying the tax to the original $80 instead of the discounted $60 is the single most common wrong-answer trap built into these items — GED test writers deliberately include a distractor option that results from doing the steps in the wrong order.
Evaluate: −3/5 + 2/3
A contractor charges a $75 flat inspection fee plus $32 per hour of labor. If a job takes 4.5 hours, what is the total charge?
Simplify: 2 1/4 ÷ 3/8
Key Takeaways
- Multiplying/dividing integers follows same-sign/different-sign rules, but adding/subtracting integers depends on magnitude, not just sign — treat these as two separate skills.
- Fraction division always uses Keep-Change-Flip; always convert mixed numbers to improper fractions before multiplying or dividing them.
- PEMDAS governs every multi-step expression — do parentheses and exponents before multiplication/division, and multiplication/division before addition/subtraction.
- In multi-step real-world problems, the order operations are applied in changes the answer — discounts, taxes, and fees are almost always sequential, not simultaneous.
- Because this domain anchors the no-calculator section, practice fraction and integer arithmetic by hand until it is fast and automatic, not just on the onscreen calculator.