Perform Operations With Multi-Digit Decimals - 6th Grade Math

A swimming pool is being filled with water. After 2.5 hours, the pool contains 847.5 gallons. If water flows at a constant rate, how many gallons will be in the pool after 6.25 hours of filling?
Answer: 2,118.75 gallons after 6.25 hours of filling This is a constant rate problem, which means you need to find how fast water flows per hour, then use that rate to calculate the total after a different amount of time. First, find the rate of water flow. Since the pool contains 847.5 gallons after 2.5 hours of constant filling, divide the total gallons by the time: $847.5 ÷ 2.5 = 339$ gallons per hour. Now multiply this rate by the new time period: $339 × 6.25 = 2,118.75$ gallons. This confirms answer C is correct. Let's examine why the other answers are wrong. Answer A (1,983.25 gallons) comes from incorrectly calculating the rate as 339 gallons per hour but then multiplying by 5.85 hours instead of 6.25 hours. Answer B (2,227.50 gallons) results from using an incorrect rate of 356.4 gallons per hour, possibly from a calculation error when dividing 847.5 by 2.5. Answer D (2,039.06 gallons) appears to stem from multiple computational mistakes, likely involving both an incorrect rate calculation and improper multiplication. When solving constant rate problems, always follow this two-step process: first find the unit rate (divide total amount by time), then multiply that rate by the new time period. Double-check your division and multiplication, especially with decimals, as small calculation errors can lead you to attractive but incorrect answer choices.
A runner jogged $18.45$ km on Saturday and $7.80$ km on Sunday. How many kilometers did the runner jog in total? (Align decimals to add.)
Answer: $26.25$ This question tests fluently adding, subtracting, multiplying, dividing multi-digit decimals using standard algorithms: align decimal points (add/subtract), count decimal places (multiply), move decimals to make whole (divide). Addition/Subtraction: align decimal points vertically (12.50 over 3.75, decimals line up), add/subtract columns right to left (5+5=10 carry, 2+7=9, 1+3=4, 0+0=0: sum 16.25), decimal in answer below operands. Multiplication: multiply ignoring decimals (3.5×2.4→35×24=840), count total decimal places in factors (3.5 has 1, 2.4 has 1, total 2), place decimal in product from right (840 with 2 places: 8.40=8.4). Division: move divisor decimal right making whole (1.2→12 move 1 place), move dividend same amount (15.6→156), divide as whole numbers (156÷12=13). For this question, add 18.45 + 7.80 by aligning decimals: 18.45 + 7.80 = 26.25. A common error is misalignment, such as adding without carrying over, resulting in 25.25. Standard algorithms: add/subtract (align decimals, operate, decimal below), multiply (ignore decimals, multiply, count places, insert decimal), divide (move decimals right, divide, quotient has decimal); verification and contexts like tracking distances help avoid mistakes like forgetting to carry.
Calculate $67.08+9.45$ . Be sure to align decimal points before adding.
Answer: $76.53$ This question tests fluently adding, subtracting, multiplying, dividing multi-digit decimals using standard algorithms: align decimal points (add/subtract), count decimal places (multiply), move decimals to make whole (divide). Addition/Subtraction: align decimal points vertically (67.08 over 9.45, decimals line up), add/subtract columns right to left (8+5=13 write 3 carry 1, 0+4+1=5, 7+9=16 write 6 carry 1, 6+0+1=7: sum 76.53), decimal in answer below operands. Multiplication: multiply ignoring decimals (3.5×2.4→35×24=840), count total decimal places in factors (3.5 has 1, 2.4 has 1, total 2), place decimal in product from right (840 with 2 places: 8.40=8.4). Division: move divisor decimal right making whole (1.2→12 move 1 place), move dividend same amount (15.6→156), divide as whole numbers (156÷12=13). For this addition, align 67.08 and 9.45, add to get 76.53. A common error is not aligning, like adding as 6708+945=7653 without decimal, or carry mistake like 76.513. Standard algorithms ensure precision: align and add with carry; verification by subtraction (76.53 - 9.45 = 67.08 checks out). Mistakes often involve alignment or arithmetic.
Verify this calculation using an inverse operation: A student says $34.70 - 18.55 = 16.15$ . Which check correctly verifies the subtraction?
Answer: Check by adding: $18.55 + 16.15 = 34.70$ This question tests fluently adding, subtracting, multiplying, and dividing multi-digit decimals using standard algorithms: align decimal points for addition and subtraction, count decimal places for multiplication, and move decimals to make the divisor whole for division, with verification using inverse operations. To verify subtraction like 34.70 - 18.55 = 16.15, use inverse addition: add the result to the subtracted number (16.15 + 18.55 = 34.70); multiplication and division have their inverses too. For example, checking 34.70 - 18.55: adding 18.55 + 16.15 gives 34.70, confirming accuracy. The correct verification is checking by adding 18.55 + 16.15 = 34.70, which is choice B. A common error is using the wrong inverse, like subtracting instead or adding incorrectly, leading to invalid checks. Standard verification: for subtraction, add back to original; this ensures reliability in calculations. Contexts include financial transactions, and mistakes often involve confusing operations or arithmetic errors.
A recipe uses $48.75$ ounces of smoothie mix to make $2.5$ equal servings. Using the standard division algorithm (move the decimal to make the divisor a whole number), how many ounces are in each serving: $48.75\div 2.5$ ?
Answer: $19.5$ This question tests fluently adding, subtracting, multiplying, dividing multi-digit decimals using standard algorithms: align decimal points (add/subtract), count decimal places (multiply), move decimals to make whole (divide). Addition/Subtraction: align decimal points vertically (12.50 over 3.75, decimals line up), add/subtract columns right to left (5+5=10 carry, 2+7=9, 1+3=4, 0+0=0: sum 16.25), decimal in answer below operands. Multiplication: multiply ignoring decimals (3.5×2.4→35×24=840), count total decimal places in factors (3.5 has 1, 2.4 has 1, total 2), place decimal in product from right (840 with 2 places: 8.40=8.4). Division: move divisor decimal right making whole (2.5→25 move 1 place), move dividend same amount (48.75→487.5), divide as whole numbers (487.5÷25=19.5). For this division, move decimals one place to get 487.5÷25=19.5 ounces per serving. A common error is moving decimals incorrectly, like 48.75÷2.5 as 1.95 or 195 without adjustment. Standard algorithms: equal decimal moves then divide; verification by multiplication (19.5 × 2.5 = 48.75 checks out). In recipe contexts, this avoids portion errors.
A student buys a notebook for $12.50 and a pack of pens for $15.75. What is the total cost? (Use the standard addition algorithm by aligning decimal points.)
Answer: $28.25 This question tests fluently adding, subtracting, multiplying, and dividing multi-digit decimals using standard algorithms: align decimal points for addition and subtraction, count decimal places for multiplication, and move decimals to make the divisor whole for division. For addition, align the decimal points vertically, such as 12.50 over 15.75, and add columns from right to left, carrying over as needed; for subtraction, align similarly and borrow when necessary; multiplication involves ignoring decimals initially, multiplying whole numbers, then placing the decimal based on total decimal places; division requires shifting decimals in both divisor and dividend to make the divisor whole, then dividing as usual. For example, adding 12.50 + 15.75: align as 12.50 and 15.75, add hundredths 0+5=5, tenths 5+7=12 (write 2, carry 1), units 2+5+1=8, tens 1+1=2, resulting in 28.25. The correct total cost is $12.50 + $15.75 = $28.25, which is choice B. A common error is misalignment, such as adding without aligning decimals leading to $282.5, or arithmetic mistakes like forgetting to carry over resulting in $27.35. Standard algorithms ensure accuracy: align decimals for addition and subtraction, operate column by column, and place the decimal directly below. Verification can be done by subtracting one addend from the sum, like $28.25 - $15.75 = $12.50, confirming the addition; this skill applies to contexts like calculating total costs in money.
A water bottle holds $45.60$ ounces when full. After practice, $12.75$ ounces are left. Using the standard subtraction algorithm (align decimal points), how many ounces were drunk?
Answer: $32.85$ This question tests fluently adding, subtracting, multiplying, dividing multi-digit decimals using standard algorithms: align decimal points (add/subtract), count decimal places (multiply), move decimals to make whole (divide). Addition/Subtraction: align decimal points vertically (45.60 over 12.75, decimals line up), add/subtract columns right to left (0-5 can't borrow to 10-5=5, 5-7 can't borrow to 15-7=8 with carryover effects, resulting in 32.85), decimal in answer below operands. Multiplication: multiply ignoring decimals (3.5×2.4→35×24=840), count total decimal places in factors (3.5 has 1, 2.4 has 1, total 2), place decimal in product from right (840 with 2 places: 8.40=8.4). Division: move divisor decimal right making whole (1.2→12 move 1 place), move dividend same amount (15.6→156), divide as whole numbers (156÷12=13). For this subtraction, align 45.60 and 12.75, subtract with borrowing to get 32.85 ounces drunk. A common error is misalignment or improper borrowing, like subtracting without borrow leading to 32.95 or arithmetic mistakes. Standard algorithms: subtract with alignment and regrouping; verification by adding back (32.85 + 12.75 = 45.60 checks out). In measurement contexts like ounces, this ensures precise calculations.
A runner jogs $26.45$ miles in one week and $18.9$ miles the next week. What is the total distance jogged? Compute $26.45 + 18.90$ by aligning decimal points.
Answer: $45.35$ This question tests fluently adding, subtracting, multiplying, and dividing multi-digit decimals using standard algorithms: align decimal points for addition and subtraction, count decimal places for multiplication, and move decimals to make the divisor whole for division. For addition, align decimals vertically, writing 26.45 + 18.90, and add from right to left with carrying; subtraction uses borrowing; multiplication ignores decimals then places them; division shifts decimals. For example, 26.45 + 18.90: align as 26.45 and 18.90, hundredths 5+0=5, tenths 4+9=13 (write 3, carry 1), units 6+8+1=15 (write 5, carry 1), tens 2+1+1=4, resulting in 45.35. The correct total distance is 26.45 + 18.9 = 45.35 miles, which is choice A. A common error is misalignment or not adding zeros, leading to 44.45 or 35.35, or carry-over mistakes like 453.5. Standard algorithms ensure reliability: align decimals, add with carrying, decimal below. Verification by subtraction, such as 45.35 - 18.9 = 26.45, works; applies to tracking distances in sports.
A recipe uses $2.75$ cups of flour per batch. You want to make $3$ batches. How many cups of flour do you need? (Compute $2.75 \times 3$ using the standard multiplication algorithm.)
Answer: $8.25$ This question tests fluently adding, subtracting, multiplying, dividing multi-digit decimals using standard algorithms: align decimal points (add/subtract), count decimal places (multiply), move decimals to make whole (divide). Addition/Subtraction: align decimal points vertically (12.50 over 3.75, decimals line up), add/subtract columns right to left (5+5=10 carry, 2+7=9, 1+3=4, 0+0=0: sum 16.25), decimal in answer below operands. Multiplication: multiply ignoring decimals (3.5×2.4→35×24=840), count total decimal places in factors (3.5 has 1, 2.4 has 1, total 2), place decimal in product from right (840 with 2 places: 8.40=8.4). Division: move divisor decimal right making whole (1.2→12 move 1 place), move dividend same amount (15.6→156), divide as whole numbers (156÷12=13). For this question, multiply 2.75 × 3 as 275 × 3 = 825, with 2 decimal places, so 8.25. A common error is forgetting decimal places, resulting in 825 or 0.825. Standard algorithms facilitate scaling in recipes, verifiable by division, preventing placement errors.
At the school store, a notebook costs $12.50 and a pack of pens costs $15.75. What is the total cost? (Use the standard addition algorithm by aligning decimal points.)
Answer: $28.25 This question tests fluently adding, subtracting, multiplying, dividing multi-digit decimals using standard algorithms: align decimal points (add/subtract), count decimal places (multiply), move decimals to make whole (divide). Addition/Subtraction: align decimal points vertically ( $12.50$ over $3.75$ , decimals line up), add/subtract columns right to left ( $5+5=10$ carry, $2+7=9$ , $1+3=4$ , $0+0=0$ : sum $16.25$ ), decimal in answer below operands. Multiplication: multiply ignoring decimals ( $3.5 \times 2.4 \to 35 \times 24 = 840$ ), count total decimal places in factors ( $3.5$ has 1, $2.4$ has 1, total 2), place decimal in product from right ( $840$ with 2 places: $8.40=8.4$ ). Division: move divisor decimal right making whole ( $1.2 \to 12$ move 1 place), move dividend same amount ( $15.6 \to 156$ ), divide as whole numbers ( $156 \div 12=13$ ). For this question, add $12.50 + 15.75$ by aligning decimals: $12.50 + 15.75 = 28.25$ . A common error is not aligning decimals properly, leading to results like 2815 or 28.15. Standard algorithms ensure accuracy in contexts like calculating total costs in money, and you can verify by subtracting one addend from the sum to get the other.
A $48.6$ -meter rope is cut into pieces that are each $1.8$ meters long. How many pieces can be cut? Compute $48.6 \div 1.8$ by moving decimals to make the divisor a whole number.
Answer: $27$ This question tests fluently adding, subtracting, multiplying, and dividing multi-digit decimals using standard algorithms: align decimal points for addition and subtraction, count decimal places for multiplication, and move decimals to make the divisor whole for division. For division, shift divisor to whole (1.8→18, 1 place), dividend accordingly (48.6→486), divide (486÷18=27); addition/subtraction align; multiplication counts places. For example, 48.6÷1.8: 486÷18=27. The correct number of pieces is 48.6 ÷ 1.8 = 27, which is choice B. A common error is unequal decimal shifts, leading to 2.7 or 270, or division mistake like 486÷18=26. Standard algorithms: move decimals right, divide whole, decimal in quotient. Verification by multiplication, 27 × 1.8 = 48.6, confirms; useful for dividing materials like ropes.
Calculate: $12.6 \times 4.5$ . (Multiply as whole numbers, then place the decimal using the total number of decimal places.)
Answer: $56.7$ This question tests fluently adding, subtracting, multiplying, dividing multi-digit decimals using standard algorithms: align decimal points (add/subtract), count decimal places (multiply), move decimals to make whole (divide). Addition/Subtraction: align decimal points vertically (12.50 over 3.75, decimals line up), add/subtract columns right to left (5+5=10 carry, 2+7=9, 1+3=4, 0+0=0: sum 16.25), decimal in answer below operands. Multiplication: multiply ignoring decimals (3.5×2.4→35×24=840), count total decimal places in factors (3.5 has 1, 2.4 has 1, total 2), place decimal in product from right (840 with 2 places: 8.40=8.4). Division: move divisor decimal right making whole (1.2→12 move 1 place), move dividend same amount (15.6→156), divide as whole numbers (156÷12=13). For this question, multiply 12.6 × 4.5 as 126 × 45 = 5670, with 2 decimal places total, so 56.70 or 56.7. A common error is wrong decimal count, like placing it to get 5.67 or 567. Standard algorithms and verification by division support accurate computations in real-world scaling problems.
A class is making bracelets. Each bracelet uses $12.5$ cm of string, and they make $3.2$ bracelets' worth of string into one long piece for a project. What is the total length of string needed? Compute $12.5 \times 3.2$ using the standard algorithm.
Answer: $40.0$ This question tests fluently adding, subtracting, multiplying, and dividing multi-digit decimals using standard algorithms: align decimal points for addition and subtraction, count decimal places for multiplication, and move decimals to make the divisor whole for division. For multiplication, ignore decimals (12.5 as 125, 3.2 as 32, 125×32=4000), count total places (1+1=2), place decimal (40.00 or 40.0); addition/subtraction align; division shifts. For example, 12.5×3.2: 125×32=4000, 2 places, 40.00=40.0. The correct total length is 12.5 × 3.2 = 40.0 cm, which is choice B. A common error is wrong decimal count, like 1 place for 400 or 3 for 4.0, or multiplication error like 125×30=3750 only. Standard algorithms: ignore decimals, multiply, insert based on count. Verification by division, 40.0 ÷ 3.2 = 12.5, checks; applies to crafts like string lengths.
A rectangular garden measures 12.8 meters by 9.45 meters. If fencing costs $7.50 per meter, what is the total cost to fence the entire perimeter of the garden?
Answer: $333.75 for complete perimeter fencing Perimeter = 2(length + width) = 2(12.8 + 9.45) = 2(22.25) = 44.5 meters. Cost = 44.5 × $7.50 = $333.75. Choice B is half the correct answer (forgetting to double for both lengths and widths). Choice C is double the correct answer. Choice D results from calculating area instead of perimeter.
Calculate: $3.5 \times 2.4$ . (Multiply as whole numbers first, then place the decimal by counting total decimal places.)
Answer: $8.4$ This question tests fluently adding, subtracting, multiplying, dividing multi-digit decimals using standard algorithms: align decimal points (add/subtract), count decimal places (multiply), move decimals to make whole (divide). Addition/Subtraction: align decimal points vertically (12.50 over 3.75, decimals line up), add/subtract columns right to left (5+5=10 carry, 2+7=9, 1+3=4, 0+0=0: sum 16.25), decimal in answer below operands. Multiplication: multiply ignoring decimals (3.5×2.4→35×24=840), count total decimal places in factors (3.5 has 1, 2.4 has 1, total 2), place decimal in product from right (840 with 2 places: 8.40=8.4). Division: move divisor decimal right making whole (1.2→12 move 1 place), move dividend same amount (15.6→156), divide as whole numbers (156÷12=13). For this question, multiply 3.5 × 2.4 as 35 × 24 = 840, with 2 decimal places total, so 8.4. A common error is miscounting decimal places, resulting in 84.0 or 0.84. Standard algorithms ensure precision, and verification by division (like 8.4 ÷ 2.4 = 3.5) confirms results in contexts like area calculations with decimal measurements.
A student buys a notebook for $12.50 and a pack of pens for $15.75. Using the standard addition algorithm (align decimal points), what is the total cost?
Answer: $28.25 This question tests fluently adding, subtracting, multiplying, dividing multi-digit decimals using standard algorithms: align decimal points (add/subtract), count decimal places (multiply), move decimals to make whole (divide). Addition/Subtraction: align decimal points vertically (12.50 over 15.75, decimals line up), add/subtract columns right to left (0+5=5, 5+7=12 write 2 carry 1, 2+5+1=8, 1+1+0=2, 0+0=0: sum 28.25), decimal in answer below operands. Multiplication: multiply ignoring decimals (3.5×2.4→35×24=840), count total decimal places in factors (3.5 has 1, 2.4 has 1, total 2), place decimal in product from right (840 with 2 places: 8.40=8.4). Division: move divisor decimal right making whole (1.2→12 move 1 place), move dividend same amount (15.6→156), divide as whole numbers (156÷12=13). For this addition, align 12.50 and 15.75, add to get 28.25, which is the total cost. A common error is not aligning decimals, like adding 1250 + 1575 as 2825 without decimal, resulting in $2825 or $28.25 misplaced. Standard algorithms ensure accuracy: for addition, align and add with carry; verification by subtracting one addend from sum (28.25 - 15.75 = 12.50 checks out). In money contexts, this prevents mistakes in totals.
Verify the sum using the inverse operation: If $27.35 + 14.80 = 42.15$ , which subtraction correctly checks the addition?
Answer: $42.15 - 14.80 = 27.35$ This question tests fluently adding, subtracting, multiplying, dividing multi-digit decimals using standard algorithms: align decimal points (add/subtract), count decimal places (multiply), move decimals to make whole (divide). Addition/Subtraction: align decimal points vertically (12.50 over 3.75, decimals line up), add/subtract columns right to left (5+5=10 carry, 2+7=9, 1+3=4, 0+0=0: sum 16.25), decimal in answer below operands. Multiplication: multiply ignoring decimals (3.5×2.4→35×24=840), count total decimal places in factors (3.5 has 1, 2.4 has 1, total 2), place decimal in product from right (840 with 2 places: 8.40=8.4). Division: move divisor decimal right making whole (1.2→12 move 1 place), move dividend same amount (15.6→156), divide as whole numbers (156÷12=13). For this question, to verify 27.35 + 14.80 = 42.15, use subtraction: 42.15 - 14.80 = 27.35 confirms it. A common error is using the wrong subtrahend or misaligning, leading to incorrect checks like 14.90. Standard algorithms and inverse operations ensure reliability in financial or measurement verifications, avoiding arithmetic mistakes.
A recipe calls for 2.75 cups of flour. If Sarah wants to make 1.5 times the recipe but only has 3.8 cups of flour available, how much more flour does she need to buy?
Answer: 0.325 cups of additional flour needed First, calculate flour needed for 1.5 times the recipe: 2.75 × 1.5 = 4.125 cups. Then find how much more is needed: 4.125 - 3.8 = 0.325 cups. Choice B results from calculating 2.75 + 1.5 - 3.8 instead of multiplying. Choice C comes from miscalculating 2.75 × 1.5 as 3.025. Choice D results from adding instead of subtracting: 4.125 - 3.8 calculated incorrectly.
Calculate $12.5+3.75$ using the standard addition algorithm (write $12.5$ as $12.50$ to align decimal places).
Answer: $16.25$ This question tests fluently adding, subtracting, multiplying, dividing multi-digit decimals using standard algorithms: align decimal points (add/subtract), count decimal places (multiply), move decimals to make whole (divide). Addition/Subtraction: align decimal points vertically (12.50 over 3.75, decimals line up), add/subtract columns right to left (0+5=5, 5+7=12 write 2 carry 1, 2+3+1=6, 1+0=1: sum 16.25), decimal in answer below operands. Multiplication: multiply ignoring decimals (3.5×2.4→35×24=840), count total decimal places in factors (3.5 has 1, 2.4 has 1, total 2), place decimal in product from right (840 with 2 places: 8.40=8.4). Division: move divisor decimal right making whole (1.2→12 move 1 place), move dividend same amount (15.6→156), divide as whole numbers (156÷12=13). For this addition, write 12.5 as 12.50, align with 3.75, add to get 16.25. A common error is not padding zeros, like adding 12.5 + 3.75 as 16.205 misplaced or 162.5 ignoring decimal. Standard algorithms: align with equivalent decimals and add; verification by subtraction (16.25 - 3.75 = 12.5 checks out). Contexts include measurements where precision matters.
A science beaker contains $72.08$ mL of solution. A student pours out $19.6$ mL. How much solution remains? Compute $72.08 - 19.60$ using the standard subtraction algorithm.
Answer: $52.48$ This question tests fluently adding, subtracting, multiplying, and dividing multi-digit decimals using standard algorithms: align decimal points for addition and subtraction, count decimal places for multiplication, and move decimals to make the divisor whole for division. For subtraction, align decimals, writing 72.08 - 19.60, and subtract with borrowing; addition aligns for carrying; multiplication counts places; division shifts to whole. For example, 72.08 - 19.60: align, hundredths 8-0=8, tenths 0-6 requires borrowing (10-6=4, units 1 becomes 0, but from 2), actually full: borrow across, resulting in 52.48. The correct remaining solution is 72.08 - 19.6 = 52.48 mL, which is choice A. A common error is borrowing mistakes, leading to 53.48 or 51.48, or misalignment like 524.8. Standard algorithms: align, subtract with borrowing, decimal in place. Verification by addition, 52.48 + 19.6 = 72.08, confirms; used in science for volumes.

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