Solve multistep rational-number problems and check interpretations

Learning goal: Choose and carry out operations with positive and negative fractions and decimals in multistep problems. Interpret complex fractions as division, use efficient number forms and verify that an answer matches the requested quantity.

A multistep problem requires a mathematical representation before calculation. Identify starting quantities, changes, repeated effects and the final question. Units and signs help determine operations. A negative value may represent a position, a decrease or an amount owed in a fictional record; those roles are not interchangeable. State the convention used in the problem.

Consider a model temperature starting at −2.5 degrees. It rises by 3/4 degree in each of four equal periods, then falls by 1.2 degrees. The total rise is 4 × 3/4 = three degrees. The ending reading is −2.5 + 3 − 1.2 = −0.7 degree. The net change is +1.8 degrees, found from three minus 1.2. The final reading and net change differ because the starting reading is not zero.

Choose a number form that simplifies rather than obscures the calculation. Here 3/4 multiplied by four gives an exact whole number, so converting it to a decimal first is unnecessary. For −1/2 + 0.35, decimals are convenient: −0.50 + 0.35 = −0.15. Fractions also work: −10/20 + 7/20 = −3/20. Both exact forms represent the same number.

A complex fraction is a fraction whose numerator, denominator or both contain a fraction, possibly inside a longer expression. It represents division of the entire numerator by the entire denominator. For [1/2 − 3/4] ÷ [−2/3], first compute the numerator: 2/4 − 3/4 = −1/4. Then divide by −2/3: (−1/4) × (−3/2) = 3/8. Brackets show the grouping and prevent an operation from being applied to only part of a quantity.

A denominator must be evaluated before confirming that division is defined. The expression [2/3]/[1/2 − 1/2] has a zero denominator and is undefined. Canceling one of the halves before evaluating the difference would destroy the expression's structure. Cancellation concerns common nonzero factors in products, not arbitrary matching numbers separated by addition or subtraction.

Use strategic grouping for longer expressions. Suppose a fictional event balance starts at 18.50 units, receives four contributions of 2.75 each and pays three costs of 6.40 each. The expression is 18.50 + 4(2.75) − 3(6.40). Contributions total eleven and costs total 19.20, giving 10.30 remaining. Recording costs as three signed changes of −6.40 gives the equivalent expression 18.50 + 11 + 3(−6.40).

Estimate independently. The balance is about 19 + 11 − 19 = eleven, so 10.30 is plausible. An answer of 48.70 likely added the costs instead of subtracting them. An estimate should be close enough to detect an error while remaining simpler than the exact computation. Merely rounding the already calculated answer is not an independent check of the method.

A measurement context can require a division after a difference. A centered strip is 9 3/4 inches long on a panel 27 1/2 inches wide. The unused width is 27 1/2 − 9 3/4 = 17 3/4 inches. Dividing that evenly between two sides gives 8 7/8 inches on each side. Half the panel width alone, 13 3/4 inches, is not the margin, because it still contains half of the strip. You may divide first if you then subtract half the strip: 13 3/4 − 4 7/8 = 8 7/8 inches. Either order is valid; what matters is that the strip is accounted for.

The panel calculation permits an estimate: about 28 minus about ten is about eighteen, and half is about nine inches. The exact 8 7/8 is consistent. Check by rebuilding the whole: 8 7/8 + 9 3/4 + 8 7/8 = 27 1/2. Every unoccupied margin belongs to the whole panel; it is not ignored because only the central strip is highlighted.

Tools can help when arithmetic is lengthy, but choose them strategically. A calculator can verify a complex decimal calculation after you write the expression with correct grouping. It cannot decide whether a cost should be subtracted or which denominator a rate requires. Record enough of the setup that another reader can evaluate the reasoning without trusting an unexplained display.

Worked compound fraction: [−1/3 + 5/6]/[3/4] has numerator −2/6 + 5/6 = 3/6 = 1/2. Divide by 3/4 to get (1/2)(4/3) = 2/3. Check by multiplying 2/3 by the denominator 3/4, which returns 1/2. This check confirms the division step only; the numerator came from the earlier common-denominator step and needs its own check.

Worked temperature interpretation: Ending at −0.7 after starting at −2.5 means a positive net change of 1.8. A negative final reading does not imply the overall change was negative. Compare final minus initial to verify: −0.7 − (−2.5) = 1.8.

Worked exact margin: Write the panel and strip in quarters: 27 1/2 = 110/4 and 9 3/4 = 39/4. Their difference is 71/4. Half is 71/8 = 8 7/8. Fractions retain the exact eighth-inch result without a rounding step.

Worked form choice: A fictional hourly rate of 24 budget units rises by 10%. Because 10% equals 1/10, the increase is 24 ÷ 10 = 2.4 units and the new rate is 26.4 units. Converting the percent to a fraction turns the step into a mental division. Estimate to check: 24 is a little less than 25, so one tenth of it is a little less than 2.5, and the new rate should be a little less than 24 + 2.5 = 26.5. The result 26.4 fits.

Planning rehearsal: For the temperature, event balance and centered strip, label the requested quantity before writing an expression. Explain the operation represented by each term. Solve in an efficient form and use a separate estimate or reverse calculation to check. Identify the whole in the centered-strip diagram and show that both margins are included.

Independent performance: Construct four multistep problems: one with signed changes, one with repeated gains and costs, one involving fractional measurement and one complex fraction with nonzero denominator. Supply sufficient context and units, solve exactly where possible and state any rounding rule. Add a fifth expression with a denominator that becomes zero and explain why it is undefined. Exchange work with a partner who checks grouping, signs, units and whether the answer matches the question.