Add, subtract, multiply and divide rational numbers

1. Rewrite "minus a negative" as "plus": -8 + 5 - (-6) = -8 + 5 + 6.
2. Work left to right. -8 + 5: the signs are different, so subtract 8 - 5 = 3 and keep the sign of -8: the result is -3.
3. -3 + 6: the signs are different, so subtract 6 - 3 = 3 and keep the sign of 6: the result is 3.

Answer: 3

Check: On a number line, start at -8, move 5 to the right (to -3), then 6 more to the right (to 3). ✓

Tip: Two negative signs side by side, -(-6), always become +6. Rewrite them first, before doing any other arithmetic.

1. There are no parentheses or exponents, so do multiplication and division first.
2. 4 \times 3 = 12 and 6 \div 2 = 3. The expression becomes 18 - 12 + 3.
3. Addition and subtraction go left to right: 18 - 12 = 6.
4. Then 6 + 3 = 9.

Answer: 9

Check: Type 18 − 4 × 3 + 6 ÷ 2 into the TI-30XS; it follows the order of operations and shows 9 ✓.

Tip: "PEMDAS" does not mean addition before subtraction. Addition and subtraction are a team: do them in the order they appear, left to right. The same goes for multiplication and division.

1. Rewrite the double signs. Adding a negative is subtracting: +(-3) = -3. Subtracting a negative is adding: -(-15) = +15.
2. The expression becomes -12 - 7 - 3 + 15.
3. Combine left to right: -12 - 7 = -19.
4. -19 - 3 = -22.
5. -22 + 15: the signs are different, so subtract 22 - 15 = 7 and keep the sign of the larger size, -22. The result is -7.

Answer: -7

Check: Group the negatives and positives: negatives -12 - 7 - 3 = -22; positives +15. Then -22 + 15 = -7 ✓.

Tip: A fast, safe method is to add up all the negative parts, add up all the positive parts, then combine the two totals once.

1. Below sea level is negative, so the start is -282 feet.
2. Climbing 1,150 feet means add: -282 + 1{,}150. The signs are different, so subtract the sizes 1{,}150 - 282 = 868 and keep the sign of the larger size (positive): 868.
3. Walking down 395 feet means subtract: 868 - 395 = 473.

Answer: 473 feet above sea level

Check: Work backwards: 473 + 395 = 868 (undo the walk down), then 868 - 1{,}150 = -282 (undo the climb) — the starting elevation ✓.

Tip: Draw a quick vertical number line with sea level at 0. Up is +, down is -. This makes it clear that starting below zero eats up part of the climb.

1. Sign: a positive times a negative is negative.
2. Cancel before multiplying: 5 and 10 share a factor of 5 (5 \to 1, 10 \to 2); 9 and 6 share a factor of 3 (9 \to 3, 6 \to 2).
3. Multiply what is left: \dfrac{1 \times 3}{2 \times 2} = \dfrac{3}{4}.
4. Attach the sign: -\dfrac{3}{4}.

Answer: -\dfrac{3}{4}

Check: Without canceling: \dfrac{5 \times 9}{6 \times 10} = \dfrac{45}{60}, and 45 \div 15 = 3, 60 \div 15 = 4, so \dfrac{45}{60} = \dfrac{3}{4}; with the negative sign, -\dfrac{3}{4} ✓.

Tip: To multiply fractions you do not need a common denominator — just multiply straight across. Common denominators are for adding and subtracting.

1. Total cost — line up the decimal points: 4.79 + 12.06 + 0.85 4.79 + 12.06 = 16.85, then 16.85 + 0.85 = 17.70. The total is \$17.70.
2. Change = amount paid - total: \$20.00 - \$17.70 = \$2.30.

Answer: $2.30

Check: Add the change back to the total: \$17.70 + \$2.30 = \$20.00 ✓. Estimate: about \$5 + \$12 + \$1 = \$18, so the change should be about \$2 ✓.

Tip: Write whole-dollar amounts with two decimal places (\$20.00) so the columns line up when you subtract.

1. Keep, change, flip: keep -\tfrac{2}{3}, change \div to \times, flip \tfrac{4}{9} to \tfrac{9}{4}: -\frac{2}{3} \div \frac{4}{9} = -\frac{2}{3} \times \frac{9}{4}
2. Multiply straight across: -\dfrac{2 \times 9}{3 \times 4} = -\dfrac{18}{12}.
3. Simplify by dividing top and bottom by 6: -\dfrac{18}{12} = -\dfrac{3}{2}.
4. Sign: a negative divided by a positive is negative.

Answer: -\tfrac{3}{2} (which is -1\tfrac{1}{2} or -1.5)

Check: Multiply back: -\tfrac{3}{2} \times \tfrac{4}{9} = -\tfrac{12}{18} = -\tfrac{2}{3} ✓.

Tip: Flip only the second fraction (the divisor). Cancel before multiplying to keep the numbers small: \tfrac{2}{3} \times \tfrac{9}{4} → cancel 3 into 9 and 2 into 4 → \tfrac{1 \times 3}{1 \times 2} = \tfrac{3}{2}.

1. "Remaining" means subtract: 8\tfrac{1}{4} - 3\tfrac{5}{6}.
2. Common denominator 12: 8\tfrac{1}{4} = 8\tfrac{3}{12} and 3\tfrac{5}{6} = 3\tfrac{10}{12}.
3. \tfrac{3}{12} is smaller than \tfrac{10}{12}, so borrow 1 from the 8: 8\tfrac{3}{12} = 7 + 1 + \tfrac{3}{12} = 7\tfrac{15}{12}.
4. Subtract: whole parts 7 - 3 = 4; fraction parts \tfrac{15}{12} - \tfrac{10}{12} = \tfrac{5}{12}.

Answer: 4\tfrac{5}{12} feet

Check: Add back: 4\tfrac{5}{12} + 3\tfrac{10}{12} = 7\tfrac{15}{12} = 8\tfrac{3}{12} = 8\tfrac{1}{4} ✓. Estimate: 8.25 - 3.83 \approx 4.4, and 4\tfrac{5}{12} \approx 4.42 ✓.

Tip: Another way that never needs borrowing: change both to improper fractions. \tfrac{33}{4} - \tfrac{23}{6} = \tfrac{99}{12} - \tfrac{46}{12} = \tfrac{53}{12} = 4\tfrac{5}{12}.

1. Start at 7^\circF.
2. A drop of 15^\circ means subtract: 7 - 15 = -8. The noon temperature was -8^\circF (below zero).
3. A rise of 9^\circ means add: -8 + 9 = 1. Different signs: 9 - 8 = 1, keep the sign of 9 (positive).

Answer: 1^\circ\text{F}

Check: The overall change is -15 + 9 = -6, and 7 - 6 = 1 ✓.

Tip: Turn words into signs: "dropped", "fell", "lost", "withdrew", "owed" → negative; "rose", "gained", "deposited", "earned" → positive. Then just add the signed numbers.

1. Write the equation: n \times \left(-\dfrac{2}{5}\right) = 8.
2. Undo multiplication by dividing both sides by -\dfrac{2}{5}: n = 8 \div \left(-\dfrac{2}{5}\right).
3. Dividing by a fraction means multiplying by its reciprocal: n = 8 \times \left(-\dfrac{5}{2}\right).
4. 8 \times 5 = 40 and 40 \div 2 = 20. A positive times a negative is negative: n = -20.

Answer: -20

Check: Put it back in: -20 \times \left(-\dfrac{2}{5}\right) = \dfrac{40}{5} = 8 ✓ (negative times negative is positive).

Tip: In "work backwards" questions, always plug your answer into the original sentence. It takes ten seconds and catches sign mistakes.

1. Step 1 is correct: multiples of 4 are 4, 8, 12; multiples of 6 are 6, 12. The LCD is 12.
2. Step 2 is correct: \tfrac{3}{4} = \tfrac{3 \times 3}{4 \times 3} = \tfrac{9}{12} and \tfrac{5}{6} = \tfrac{5 \times 2}{6 \times 2} = \tfrac{10}{12}.
3. Step 3 is wrong. The two numbers have different signs, so you subtract the sizes, 10 - 9 = 1, and keep the sign of the larger size (+\tfrac{10}{12}). The correct sum is +\tfrac{1}{12}. Maria added the sizes (9 + 10 = 19) and kept the negative sign.
4. Step 4 correctly rewrites -\tfrac{19}{12}, but it is built on the wrong number from Step 3.

Answer: Step 3

Check: \tfrac{5}{6} \approx 0.83 is a bit more than \tfrac{3}{4} = 0.75, so the sum should be a small positive number: 0.83 - 0.75 \approx 0.08 \approx \tfrac{1}{12} ✓.

Tip: Same signs → add the sizes, keep the sign. Different signs → subtract the sizes, keep the sign of the bigger one. This works for fractions exactly as for integers.

1. "How many bows" = total length \div length per bow: 20\tfrac{1}{4} \div 1\tfrac{1}{8}.
2. Improper fractions: 20\tfrac{1}{4} = \tfrac{81}{4} and 1\tfrac{1}{8} = \tfrac{9}{8}.
3. Multiply by the reciprocal: \tfrac{81}{4} \times \tfrac{8}{9}.
4. Cancel: 81 \div 9 = 9 and 8 \div 4 = 2. So 9 \times 2 = 18.

Answer: 18 bows

Check: 18 \times 1\tfrac{1}{8} = 18 + \tfrac{18}{8} = 18 + 2\tfrac{1}{4} = 20\tfrac{1}{4} yards ✓ — exactly the whole roll.

Tip: You cannot divide mixed numbers part by part. Always convert to improper fractions (or decimals: 20.25 \div 1.125 = 18) before dividing.

1. Exponent: \left(-\tfrac{1}{2}\right)^2 = \left(-\tfrac{1}{2}\right)\left(-\tfrac{1}{2}\right) = \tfrac{1}{4} (negative times negative is positive).
2. Division: \tfrac{1}{4} \div \tfrac{1}{8} = \tfrac{1}{4} \times \tfrac{8}{1} = 2.
3. Multiplication: 3 \times \left(-\tfrac{2}{3}\right) = -2.
4. Subtraction: 2 - (-2) = 2 + 2 = 4.

Answer: 4

Check: Ask "how many eighths are in one quarter?" Two — so step 2 gives 2 ✓. Then subtracting -2 adds 2 ✓.

Tip: Do each "chunk" separately (the part before the minus sign and the part after it), write each result down, then combine. Chunking prevents left-to-right mistakes.

1. Parentheses: \tfrac{5}{2} - 4 = \tfrac{5}{2} - \tfrac{8}{2} = -\tfrac{3}{2}.
2. Exponent: \left(-\tfrac{3}{2}\right)^2 = \left(-\tfrac{3}{2}\right)\left(-\tfrac{3}{2}\right) = \tfrac{9}{4} (negative × negative = positive).
3. Multiply: \tfrac{1}{2} \times \tfrac{9}{4} = \tfrac{9}{8}.
4. Subtract: \tfrac{3}{4} - \tfrac{9}{8} = \tfrac{6}{8} - \tfrac{9}{8} = -\tfrac{3}{8}.

Answer: -\tfrac{3}{8}

Check: With decimals, 2.5 - 4 = -1.5; (-1.5)^2 = 2.25; 0.5 \times 2.25 = 1.125; 0.75 - 1.125 = -0.375 = -\tfrac{3}{8} ✓.

Tip: The number in front of the parentheses (here \tfrac{1}{2}) multiplies the result of the exponent. The exponent is done before the multiplication, and the subtraction is done last.

1. Step 1 is correct. Division and multiplication are done left to right, and 8 \div 4 comes first.
2. Step 2 is correct. A positive times a negative is negative: 2 \times (-3) = -6.
3. Step 3 is wrong. Subtracting a negative means adding: 24 - (-6) = 24 + 6 = 30, not 18. Jaylen treated -(-6) as -6.
4. Step 4 adds correctly, but uses the wrong number from Step 3. The correct final answer is 30 + 1 = 31.

Answer: Step 3

Check: Evaluate the whole thing correctly: 24 - 2 \times (-3) + 1 = 24 + 6 + 1 = 31. Jaylen got 19, so an error happened — and the first step that disagrees with the correct path is Step 3 ✓.

Tip: In "find the error" questions, redo the problem yourself first, then compare line by line. The first line that differs from yours is the answer — later lines can look right and still carry the error.

1. "\tfrac{3}{4} of 2\tfrac{2}{3}" means \tfrac{3}{4} \times 2\tfrac{2}{3}.
2. Change the mixed number to an improper fraction: 2\tfrac{2}{3} = \tfrac{2 \times 3 + 2}{3} = \tfrac{8}{3}.
3. Multiply: \tfrac{3}{4} \times \tfrac{8}{3}. Cancel the 3s, and 8 \div 4 = 2: the product is \tfrac{2}{1} = 2.

Answer: 2 cups

Check: Making less than the full recipe should need less than 2\tfrac{2}{3} cups but more than half of it (1\tfrac{1}{3}). 2 cups fits ✓. Also, 2 \div \tfrac{3}{4} = \tfrac{8}{3} = 2\tfrac{2}{3}, the full amount ✓.

Tip: Scaling a recipe up or down is always multiplication by the scale factor. A factor less than 1 makes the amount smaller.

1. Numerator: multiply before adding: 7 \times (-2) = -14, so -3 + (-14) = -17.
2. Denominator: \tfrac{1}{2} - 2\tfrac{1}{4} = \tfrac{2}{4} - \tfrac{9}{4} = -\tfrac{7}{4}.
3. Divide: -17 \div \left(-\tfrac{7}{4}\right) = -17 \times \left(-\tfrac{4}{7}\right).
4. Negative times negative is positive: \tfrac{17 \times 4}{7} = \tfrac{68}{7} = 9\tfrac{5}{7}.

Answer: \dfrac{68}{7}

Check: Decimals: numerator -17, denominator 0.5 - 2.25 = -1.75; -17 \div -1.75 \approx 9.714, and 68 \div 7 \approx 9.714 ✓.

Tip: Treat a big fraction bar as if there were parentheses around the whole top and the whole bottom. On the calculator, type (−3+7×(−2))÷(1/2−9/4) with both sets of parentheses.

1. Write the steps as an equation: -1\tfrac{1}{2}\,n + 7 = -2, or -\tfrac{3}{2}n + 7 = -2.
2. Undo "add 7" (the last step) by subtracting 7: -\tfrac{3}{2}n = -2 - 7 = -9.
3. Undo "multiply by -\tfrac{3}{2}" by dividing by -\tfrac{3}{2}: n = -9 \div \left(-\tfrac{3}{2}\right) = -9 \times \left(-\tfrac{2}{3}\right).
4. Negative times negative is positive: \tfrac{9 \times 2}{3} = 6.

Answer: 6

Check: Run it forward: 6 \times \left(-1\tfrac{1}{2}\right) = -9, then -9 + 7 = -2 ✓.

Tip: Working backwards is like taking off socks and shoes: you undo the last thing first. Each undo uses the opposite operation.

1. Step 1 is correct: 3\tfrac{1}{3} = \tfrac{3 \times 3 + 1}{3} = \tfrac{10}{3} and 1\tfrac{1}{4} = \tfrac{1 \times 4 + 1}{4} = \tfrac{5}{4}.
2. Step 2 is wrong. To divide, keep the first fraction, change \div to \times, and flip the second fraction (the divisor): \tfrac{10}{3} \times \tfrac{4}{5}. Ana flipped the first fraction instead.
3. Steps 3 and 4 multiply and simplify correctly (\tfrac{15}{40} = \tfrac{3}{8}), but they continue from the wrong Step 2.
4. The correct result: \tfrac{10}{3} \times \tfrac{4}{5} = \tfrac{40}{15} = \tfrac{8}{3} = 2\tfrac{2}{3}.

Answer: Step 2

Check: Estimate: 3\tfrac{1}{3} \div 1\tfrac{1}{4} is roughly 3 \div 1, so the answer should be a bit less than 3 — like 2\tfrac{2}{3}. Ana's \tfrac{3}{8} is far too small, a sign that something went wrong early ✓.

Tip: Remember "Keep – Change – Flip": keep the first, change the sign, flip the second. A quick estimate tells you if a division answer is in the right ballpark.

1. Total used: 12\tfrac{3}{8} + 9\tfrac{5}{6}. Whole numbers: 12 + 9 = 21. Fractions: LCD of 8 and 6 is 24, so \tfrac{3}{8} = \tfrac{9}{24} and \tfrac{5}{6} = \tfrac{20}{24}; sum = \tfrac{29}{24} = 1\tfrac{5}{24}. Total used: 21 + 1\tfrac{5}{24} = 22\tfrac{5}{24} feet.
2. Left: 30 - 22\tfrac{5}{24}. Regroup: 30 = 29\tfrac{24}{24}.
3. 29\tfrac{24}{24} - 22\tfrac{5}{24} = 7\tfrac{19}{24}.

Answer: 7\tfrac{19}{24} feet

Check: Add back: 22\tfrac{5}{24} + 7\tfrac{19}{24} = 29\tfrac{24}{24} = 30 ✓. Estimate: 12.4 + 9.8 = 22.2 and 30 - 22.2 = 7.8; 7\tfrac{19}{24} \approx 7.79 ✓.

Tip: To subtract a mixed number from a whole number, rewrite the whole number as "one less, plus a full fraction" — here 30 = 29\tfrac{24}{24}.

1. Turkey: 2.4 \times 7.85. Ignore decimals: 24 \times 785 = 18{,}840. Decimal places: 1 + 2 = 3, so 18.840 = \$18.84.
2. Cheese: 1.75 \times 6.40. Ignore decimals: 175 \times 640 = 112{,}000. Decimal places: 2 + 2 = 4, so 11.2000 = \$11.20.
3. Total before coupon: 18.84 + 11.20 = \$30.04.
4. After coupon: 30.04 - 5.00 = \$25.04.

Answer: $25.04

Check: Estimate: 2.4 \times 8 \approx 19 and 1.75 \times 6.4 \approx 11; 19 + 11 - 5 = 25 ✓.

Tip: Each item needs its own multiplication. Never add the weights together and the prices together — the items have different prices per pound.