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Ratios, Rates, and Proportions Word Problems Practice
Fifty original questions covering conversions, unit cancellation, scales, recipes, production, changing groups, dimensions, and proportional validity.
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Question 1
Explanation
Orient the factor so feet cancel: \(18\text{ feet}\cdot 12\text{ inches}/(1\text{ foot})=216\text{ inches}\). The desired unit remains.
Question 2
Explanation
Orient the factor so hours cancel: \(2.5\text{ hours}\cdot 60\text{ minutes}/(1\text{ hour})=150\text{ minutes}\). The desired unit remains.
Question 3
Explanation
Orient the factor so kilograms cancel: \(3.8\text{ kilograms}\cdot 1000\text{ grams}/(1\text{ kilogram})=3800\text{ grams}\). The desired unit remains.
Question 4
Explanation
Orient the factor so yards cancel: \(42\text{ yards}\cdot 3\text{ feet}/(1\text{ yard})=126\text{ feet}\). The desired unit remains.
Question 5
Explanation
Orient the factor so liters cancel: \(6.25\text{ liters}\cdot 1000\text{ milliliters}/(1\text{ liter})=6250\text{ milliliters}\). The desired unit remains.
Question 6
Explanation
The unit rate is \(180/12=15\) pages per minute. Multiply by \(35\) minutes: \(15(35)=525\) pages.
Question 7
Explanation
The unit rate is \(96/8=12\) boxes per hour. Multiply by \(15\) hours: \(12(15)=180\) boxes.
Question 8
Explanation
The unit rate is \(420/7=60\) labels per minute. Multiply by \(18\) minutes: \(60(18)=1080\) labels.
Question 9
Explanation
The unit rate is \(360/6=60\) kilograms per plot. Multiply by \(11\) plots: \(60(11)=660\) kilograms.
Question 10
Explanation
The unit rate is \(275/5=55\) kilometers per hour. Multiply by \(8\) hours: \(55(8)=440\) kilometers.
Question 11
Explanation
The common multiplier is \(24/3=8\). The second group has \(5(8)=40\), so the total is \(24+40=64\).
Question 12
Explanation
The common multiplier is \(28/4=7\). The second group has \(7(7)=49\), so the total is \(28+49=77\).
Question 13
Explanation
The common multiplier is \(18/2=9\). The second group has \(9(9)=81\), so the total is \(18+81=99\).
Question 14
Explanation
The common multiplier is \(40/5=8\). The second group has \(8(8)=64\), so the total is \(40+64=104\).
Question 15
Explanation
The common multiplier is \(49/7=7\). The second group has \(11(7)=77\), so the total is \(49+77=126\).
Question 16
Explanation
Keep model over actual: \(2/5=7.2/x\). Then \(2x=5(7.2)\), so \(x=18\) meters.
Question 17
Explanation
Keep model over actual: \(1.5/6=4.5/x\). Then \(1.5x=6(4.5)\), so \(x=18\) miles.
Question 18
Explanation
Keep model over actual: \(4/3=30/x\). Then \(4x=3(30)\), so \(x=22.5\) meters.
Question 19
Explanation
Keep model over actual: \(2.5/4=11.25/x\). Then \(2.5x=4(11.25)\), so \(x=18\) meters.
Question 20
Explanation
Keep model over actual: \(3/10=8.4/x\). Then \(3x=10(8.4)\), so \(x=28\) feet.
Question 21
Explanation
The scale factor is \(14/6=2.3333333333333335\). Multiply the ingredient by that same factor: \(4(2.3333333333333335)=21\).
Question 22
Explanation
The scale factor is \(25/10=2.5\). Multiply the ingredient by that same factor: \(3(2.5)=7.5\).
Question 23
Explanation
The scale factor is \(20/8=2.5\). Multiply the ingredient by that same factor: \(5(2.5)=12.5\).
Question 24
Explanation
The scale factor is \(30/12=2.5\). Multiply the ingredient by that same factor: \(2.5(2.5)=6.25\).
Question 25
Explanation
The scale factor is \(40/15=2.6666666666666665\). Multiply the ingredient by that same factor: \(1.8(2.6666666666666665)=4.8\).
Question 26
Explanation
Output per machine is \(360/4=90\). For \(7\) machines, expected output is \(90(7)=630\) panels.
Question 27
Explanation
Output per machine is \(540/6=90\). For \(10\) machines, expected output is \(90(10)=900\) booklets.
Question 28
Explanation
Output per machine is \(1040/8=130\). For \(13\) machines, expected output is \(130(13)=1690\) parts.
Question 29
Explanation
Output per machine is \(425/5=85\). For \(12\) machines, expected output is \(85(12)=1020\) trays.
Question 30
Explanation
Output per machine is \(1170/9=130\). For \(15\) machines, expected output is \(130(15)=1950\) devices.
Question 31
Explanation
Use \(72\text{ mi}/\text{h}\cdot1.6\text{ km}/\text{mi}\cdot1\text{ h}/60\text{ min}\cdot25\text{ min}=48\text{ km}\). Miles, hours, and minutes cancel to leave kilometers.
Question 32
Explanation
Convert time and distance: \(18\text{ ft}/\text{s}\cdot60\text{ s}/\text{min}\cdot2.5\text{ min}\cdot1\text{ yd}/3\text{ ft}=900\text{ yd}\).
Question 33
Explanation
Use \(45\text{ L}/\text{min}\cdot18\text{ s}\cdot1\text{ min}/60\text{ s}\cdot1000\text{ mL}/1\text{ L}=13{,}500\text{ mL}\).
Question 34
Explanation
The rate is \(36/24=1.5\) pages per minute. Convert \(1.8\) hours to \(108\) minutes, then \(1.5(108)=162\) pages.
Question 35
Explanation
Use \(2.4\text{ m}^3/\text{h}\cdot15\text{ min}\cdot1\text{ h}/60\text{ min}\cdot1000\text{ L}/1\text{ m}^3=600\text{ L}\).
Question 36
Explanation
Originally herbs \(=5k\) and flowers \(=8k\). New equality gives \(5k+21=8k\), so \(k=7\) and original total \(=13(7)=91\).
Question 37
Explanation
Adults \(=7k\), teens \(=5k\), and \(5k+12=8k\). Thus \(k=4\) and original total is \(12(4)=48\).
Question 38
Explanation
Initially A is \(3/10(50)=15\) and B is \(35\). To reach \(1:1\), A must also be \(35\), so add \(20\) kilograms.
Question 39
Explanation
Originally digital \(=9k\) and paper \(=4k\). Solve \((9k-30)/(4k)=3/2\): \(18k-60=12k\), so \(k=10\) and total \(=13k=130\).
Question 40
Explanation
Originally seniors \(=2k\) and juniors \(=5k\). Equality after the change gives \(2k+18=5k\), so \(k=6\) and original total \(=7k=42\).
Question 41
Explanation
Since \(1\text{ yd}=3\text{ ft}\), then \(1\text{ yd}^2=9\text{ ft}^2\). Therefore \(18(9)=162\) square feet.
Question 42
Explanation
The linear factor is \(15/6=2.5\) meters per centimeter, so the area factor is \(2.5^2=6.25\). Thus \(10(6.25)=62.5\) square meters.
Question 43
Explanation
Eight seconds is \(8/60\) minute, so volume is \(0.75(8/60)=0.1\) cubic meter. Since \(1\text{ m}^3=1{,}000{,}000\text{ cm}^3\), the result is \(100{,}000\) cubic centimeters.
Question 44
Explanation
The area factor is \(12^2=144\) square miles per square inch. Multiply \(2.5(144)=360\) square miles.
Question 45
Explanation
The tank receives \(240(36)=8{,}640\) cubic inches. Convert with \(1\text{ ft}^3/1{,}728\text{ in}^3\): \(8{,}640/1{,}728=5\) cubic feet.
Question 46
Explanation
A proportional relationship has form \(y=kx\) and passes through the origin. Here \(C=2.4m+9\) has a nonzero fixed charge.
Question 47
Explanation
The serving factor is \(16/12=4/3\), while the broth factor is \(9/5\). Because the factors differ, the ingredient ratio changes.
Question 48
Explanation
Output per machine-hour is \(720/(6\cdot4)=30\). Ten machines for six hours provide \(60\) machine-hours, so output is \(30(60)=1{,}800\).
Question 49
Explanation
Convert distance: \(375(1.6)=600\) kilometers. Fuel is \(8/100(600)=48\) liters, and \(48/4=12\) gallons. Each conversion leaves the requested final unit.
Question 50
Explanation
Scale factors multiply, not add. The combined area factor is \(0.8\cdot1.2=0.96\), which is less than \(1\).
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- Question 40Changing group ratiosMedium
- Question 41Area and volume conversionsHard
- Question 42Area and volume conversionsHard
- Question 43Area and volume conversionsHard
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- Question 45Area and volume conversionsHard
- Question 46Proportional validity and multi-step modelingHard
- Question 47Proportional validity and multi-step modelingHard
- Question 48Proportional validity and multi-step modelingHard
- Question 49Proportional validity and multi-step modelingHard
- Question 50Proportional validity and multi-step modelingHard