6th Grade Math Quiz: Find Volume With Fractional Edge Lengths
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Find Volume With Fractional Edge LengthsQuestion 1 of 20

A cube-shaped block has edge length 12 ft\tfrac{1}{2}\text{ ft}. What is the volume of this cube in cubic feet? (Use V=lwhV=lwh.)

16 ft3\tfrac{1}{6}\text{ ft}^3
32 ft3\tfrac{3}{2}\text{ ft}^3
18 ft\tfrac{1}{8}\text{ ft}
18 ft3\tfrac{1}{8}\text{ ft}^3
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6th Grade Math Quiz

6th Grade Math Quiz: Find Volume With Fractional Edge Lengths

Practice Find Volume With Fractional Edge Lengths in 6th Grade Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

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This quiz focuses on Find Volume With Fractional Edge Lengths, giving you a quick way to practice the rules, question types, and explanations that matter most for 6th Grade Math.

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Question 1

A cube-shaped block has edge length 12 ft\tfrac{1}{2}\text{ ft}. What is the volume of this cube in cubic feet? (Use V=lwhV=lwh.)

  1. 16 ft3\tfrac{1}{6}\text{ ft}^3
  2. 32 ft3\tfrac{3}{2}\text{ ft}^3
  3. 18 ft\tfrac{1}{8}\text{ ft}
  4. 18 ft3\tfrac{1}{8}\text{ ft}^3 (correct answer)
Explanation: Finding the volume of a cube means measuring how much space fits inside it. Since a cube has equal edges, every side here is 12\tfrac{1}{2} ft long. We use V=l×w×hV = l \times w \times h. Plug in the edge length for all three: V=12×12×12V = \tfrac{1}{2} \times \tfrac{1}{2} \times \tfrac{1}{2} Multiply the fractions by multiplying the tops and the bottoms: V=1×1×12×2×2=18 ft3V = \tfrac{1 \times 1 \times 1}{2 \times 2 \times 2} = \tfrac{1}{8} \text{ ft}^3 So the volume is 18 ft3\tfrac{1}{8}\text{ ft}^3. Here's a way to picture it: imagine slicing a 1-foot cube in half in all three directions. You'd get 88 little cubes, and our block is exactly one of those pieces — 18\tfrac{1}{8} of the whole. Try this at home: measure a small box, then multiply length × width × height to find its volume!

Question 2

A rectangular prism has dimensions 114 m×2 m×3 m1\tfrac{1}{4}\text{ m} \times 2\text{ m} \times 3\text{ m}. What is the volume in cubic meters? (Use V=lwhV=lwh.)

  1. 7.5 m37.5\text{ m}^3 (correct answer)
  2. 7.25 m37.25\text{ m}^3
  3. 6 m36\text{ m}^3
  4. 254 m3\tfrac{25}{4}\text{ m}^3
Explanation: Finding the volume of a box just means asking, "How much space fits inside?" The recipe is V=l×w×hV = l \times w \times h—multiply all three dimensions together. Let's use this prism's numbers. First, change the mixed number into a decimal: 114=1.251\tfrac{1}{4} = 1.25. Now multiply step by step. Start with two of the sides: 1.25×2=2.51.25 \times 2 = 2.5 Then multiply by the last side: 2.5×3=7.52.5 \times 3 = 7.5 So the volume is 7.5 m37.5\text{ m}^3. Think of it like stacking sugar cubes inside the box. You fill a flat layer, then stack layers up to the top. Multiplying length, width, and height counts every cube at once! Try this at home: grab a cereal box, measure its length, width, and height, and multiply them. You just found its volume!

Question 3

A sandbox is shaped like a rectangular prism. Its base is 5 m5\text{ m} by 4 m4\text{ m}, and its height is 2.5 m2.5\text{ m}. What is the volume of the sandbox in cubic meters? (Use V=bhV=bh.)

  1. 22.5 m322.5\text{ m}^3
  2. 50 m350\text{ m}^3 (correct answer)
  3. 20 m320\text{ m}^3
  4. 50 m250\text{ m}^2
Explanation: This question tests calculating the volume of rectangular prisms with fractional edge lengths using V=lwh (multiply three edges) or V=bh (base area times height), understanding packing with fractional unit cubes gives same result. Volume V=lwh: multiply length, width, height (2 ft × 3 ft × (1/2) ft = 2×3×0.5=3 ft³, fractional edges multiply like any numbers). Or V=bh: base area l×w times height (base 5×4=20 m², height 2.5 m, volume 20×2.5=50 m³). Packing: fill prism with unit cubes (if edge 1/2 ft, unit cube is (1/2)³=1/8 ft³, box 2×3×(1/2) holds 2÷(1/2)×3÷(1/2)×(1/2)÷(1/2)=4×6×1=24 cubes of 1/8 ft³ each, total 24×(1/8)=3 ft³—same as V=lwh). Units: cubic (ft³, cm³, in³—length unit cubed). For this sandbox, base area=5×4=20 m², then V=20×2.5=50 m³. Common errors include using square units (50 m²) or miscalculating base area.

Question 4

A rectangular prism has base dimensions 3.2 cm3.2\text{ cm} by 1.5 cm1.5\text{ cm} and height 4 cm4\text{ cm}. What is the volume in cubic centimeters? (Use V=bhV=bh.)

  1. 8.7 cm38.7\text{ cm}^3
  2. 4.7 cm34.7\text{ cm}^3
  3. 19.2 cm319.2\text{ cm}^3 (correct answer)
  4. 19.2 cm219.2\text{ cm}^2
Explanation: This question tests calculating the volume of rectangular prisms with fractional edge lengths using V=lwh (multiply three edges) or V=bh (base area times height), understanding packing with fractional unit cubes gives same result. Volume V=lwh: multiply length, width, height (2 ft × 3 ft × (1/2) ft = 2×3×0.5=3 ft³, fractional edges multiply like any numbers). Or V=bh: base area l×w times height (base 3.2×1.5=4.8 cm², height 4 cm, volume 4.8×4=19.2 cm³). Packing: fill prism with unit cubes (if edge 1/2 ft, unit cube is (1/2)³=1/8 ft³, box 2×3×(1/2) holds 2÷(1/2)×3÷(1/2)×(1/2)÷(1/2)=4×6×1=24 cubes of 1/8 ft³ each, total 24×(1/8)=3 ft³—same as V=lwh). Units: cubic (ft³, cm³, in³—length unit cubed). For this prism, base=3.2×1.5=4.8 cm², V=4.8×4=19.2 cm³. Common errors include decimal multiplication mistakes or using square units.

Question 5

Maria is building a concrete foundation that measures 3123\frac{1}{2} yards long, 2142\frac{1}{4} yards wide, and 23\frac{2}{3} yards thick. Concrete costs $45 per cubic yard. What is the total cost of the concrete needed for this foundation?

  1. $236.25 (correct answer)
  2. $315.00
  3. $472.50
  4. $525.00
Explanation: First convert mixed numbers: 312=723\frac{1}{2} = \frac{7}{2} and 214=942\frac{1}{4} = \frac{9}{4}. Volume = 72×94×23=7×9×22×4×3=12624=214=5.25\frac{7}{2} \times \frac{9}{4} \times \frac{2}{3} = \frac{7 \times 9 \times 2}{2 \times 4 \times 3} = \frac{126}{24} = \frac{21}{4} = 5.25 cubic yards. Cost = 5.25×45=236.255.25 \times 45 = 236.25. Choice B uses incorrect conversion of mixed numbers. Choice C doubles the correct volume. Choice D uses wrong dimension calculations.

Question 6

A science lab container is a rectangular prism that measures 2.5 cm×4 cm×3 cm2.5\text{ cm} \times 4\text{ cm} \times 3\text{ cm}. What is its volume in cubic centimeters? (Use V=lwhV=lwh.)

  1. 30 cm230\text{ cm}^2
  2. 9.5 cm39.5\text{ cm}^3
  3. 100 cm3100\text{ cm}^3
  4. 30 cm330\text{ cm}^3 (correct answer)
Explanation: This question tests calculating the volume of rectangular prisms with fractional edge lengths using V=lwh (multiply three edges) or V=bh (base area times height), understanding packing with fractional unit cubes gives same result. Volume V=lwh: multiply length, width, height (2.5 cm × 4 cm × 3 cm = 2.5×4×3=30 cm³, fractional edges multiply like any numbers). Or V=bh: base area l×w times height (base 5×4=20 m², height 2.5 m, volume 20×2.5=50 m³). Packing: fill prism with unit cubes (if edge 1/2 ft, unit cube is (1/2)³=1/8 ft³, box 2×3×(1/2) holds 2÷(1/2)×3÷(1/2)×(1/2)÷(1/2)=4×6×1=24 cubes of 1/8 ft³ each, total 24×(1/8)=3 ft³—same as V=lwh). Units: cubic (ft³, cm³, in³—length unit cubed). For this container, V=2.5×4×3=10×3=30 cm³. Common errors include multiplying only two dimensions (2.5×4=10) or using square units (30 cm²).

Question 7

A storage box is shaped like a rectangular prism with dimensions 2 ft×3 ft×12 ft2\text{ ft} \times 3\text{ ft} \times \tfrac{1}{2}\text{ ft}. What is the volume of the box in cubic feet? (Use V=lwhV=lwh.)

  1. 5.5 ft35.5\text{ ft}^3
  2. 6 ft36\text{ ft}^3
  3. 3 ft33\text{ ft}^3 (correct answer)
  4. 3 ft3\text{ ft}
Explanation: This question tests calculating the volume of rectangular prisms with fractional edge lengths using V=lwh (multiply three edges) or V=bh (base area times height), understanding packing with fractional unit cubes gives same result. Volume V=lwh: multiply length, width, height (2 ft × 3 ft × (1/2) ft = 2×3×0.5=3 ft³, fractional edges multiply like any numbers). Or V=bh: base area l×w times height (base 5×4=20 m², height 2.5 m, volume 20×2.5=50 m³). Packing: fill prism with unit cubes (if edge 1/2 ft, unit cube is (1/2)³=1/8 ft³, box 2×3×(1/2) holds 2÷(1/2)×3÷(1/2)×(1/2)÷(1/2)=4×6×1=24 cubes of 1/8 ft³ each, total 24×(1/8)=3 ft³—same as V=lwh). Units: cubic (ft³, cm³, in³—length unit cubed). For this storage box, V=2×3×(1/2)=6×0.5=3 ft³. Common errors include adding dimensions (2+3+0.5=5.5) or forgetting cubic units (3 ft).

Question 8

A rectangular prism has dimensions 34 ft×4 ft×2 ft\tfrac{3}{4}\text{ ft} \times 4\text{ ft} \times 2\text{ ft}. What is the volume of the prism in cubic feet? (Use V=lwhV=lwh.)

  1. 32 ft3\tfrac{3}{2}\text{ ft}^3
  2. 134 ft3\tfrac{13}{4}\text{ ft}^3
  3. 8 ft38\text{ ft}^3
  4. 6 ft36\text{ ft}^3 (correct answer)
Explanation: This question tests calculating the volume of rectangular prisms with fractional edge lengths using V=lwh (multiply three edges) or V=bh (base area times height), understanding packing with fractional unit cubes gives same result. Volume V=lwh: multiply length, width, height (3/4 ft × 4 ft × 2 ft = (3/4)×4×2=6 ft³, fractional edges multiply like any numbers). Or V=bh: base area l×w times height (base 5×4=20 m², height 2.5 m, volume 20×2.5=50 m³). Packing: fill prism with unit cubes (if edge 1/2 ft, unit cube is (1/2)³=1/8 ft³, box 2×3×(1/2) holds 2÷(1/2)×3÷(1/2)×(1/2)÷(1/2)=4×6×1=24 cubes of 1/8 ft³ each, total 24×(1/8)=3 ft³—same as V=lwh). Units: cubic (ft³, cm³, in³—length unit cubed). For this prism, V=(3/4)×4×2=3×2=6 ft³. Common errors include improper fraction multiplication or adding dimensions.

Question 9

A rectangular prism has a base area of 1212 cm212\tfrac{1}{2}\text{ cm}^2 and a height of 45 cm\tfrac{4}{5}\text{ cm}. What is the volume of the prism in cubic centimeters? (Use V=bhV=bh.)

  1. 10 cm310\text{ cm}^3 (correct answer)
  2. 16 cm316\text{ cm}^3
  3. 13.3 cm313.3\text{ cm}^3
  4. 10 cm210\text{ cm}^2
Explanation: Finding the volume of a prism is like stacking layers! The formula V=bhV = b h tells us to multiply the base area by the height. Let's use our numbers. First, turn the mixed number into a fraction: 1212=25212\tfrac{1}{2} = \frac{25}{2} The height is already a fraction: 45\frac{4}{5}. Now multiply across the tops and across the bottoms: V=252×45=25×42×5=10010=10V = \frac{25}{2} \times \frac{4}{5} = \frac{25 \times 4}{2 \times 5} = \frac{100}{10} = 10 Since we are measuring space inside a 3-D shape, the units are cubic centimeters: 10 cm310\text{ cm}^3. Think of the base as the floor of a box, and the height as how tall you stack that floor. Multiply them, and you fill the whole box! Try this at home: Pick a base area and a fraction height, then multiply them like fractions to find your own volume. 📦

Question 10

A rectangular prism is 1 ft1\text{ ft} long, 12 ft\tfrac{1}{2}\text{ ft} wide, and 12 ft\tfrac{1}{2}\text{ ft} tall. If you pack it completely with cubes that each have edge length 12 ft\tfrac{1}{2}\text{ ft}, how many such cubes fit inside?

  1. 2 (correct answer)
  2. 1
  3. 8
  4. 4
Explanation: To calculate the volume of a rectangular prism with fractional edge lengths, use the formula V = l × w × h, where you multiply the three dimensions, or V = base area × height, and understand that packing the prism with fractional unit cubes yields the same result. For example, with dimensions 2 ft × 3 ft × (1/2) ft, V = 2 × 3 × 0.5 = 3 ft³; alternatively, using base area, if the base is 5 m × 4 m = 20 m² and height 2.5 m, then V = 20 × 2.5 = 50 m³; packing with 1/2 ft cubes in a 2 × 3 × (1/2) prism holds 4 × 6 × 1 = 24 cubes, each 1/8 ft³, totaling 3 ft³, with units always in cubic form like ft³ though here it's count. Another example: a prism 2 ft × 3 ft × (1/2) ft has V = 2 × 3 × (1/2) = 3 ft³; or 2.5 cm × 4 cm × 3 cm: V = 30 cm³; or base 6 × 5 = 30 cm², height 2.5 cm, V = 75 cm³. For this prism 1 ft × (1/2) ft × (1/2) ft packed with (1/2) ft cubes, along length 1/(1/2)=2, width (1/2)/(1/2)=1, height (1/2)/(1/2)=1, so 2 × 1 × 1 = 2 cubes. Common errors include thinking volume is 1 × 0.5 × 0.5 = 0.25 then dividing wrong to 1 or 8, confusing with cube volume (1/2)^3=1/8 then miscounting to 4, adding, or ignoring packing. To calculate packing, divide each dimension by cube edge: 1/(0.5)=2, etc., multiply counts 2×1×1=2; relates to V / cube volume = (0.25) / (0.125) = 2. Packing verifies formula; real for storage; avoid misdividing, confusing volume with count, adding, wrong units.

Question 11

A cube-shaped block has edge length 12 ft\tfrac{1}{2}\text{ ft}. If you use cubes with edge length 12 ft\tfrac{1}{2}\text{ ft} as your unit cubes, how many unit cubes fit exactly inside the block?

  1. 4
  2. 18\tfrac{1}{8}
  3. 8
  4. 1 (correct answer)
Explanation: This question tests understanding packing rectangular prisms with fractional unit cubes, which relates to volume calculation V=lwh giving the same result. The block is a cube with edge 1/2 ft, and unit cubes are also 1/2 ft edge, so along each dimension: (1/2) ÷ (1/2) = 1, total 1×1×1=1 unit cube fits exactly. Volume-wise, V=(1/2)×(1/2)×(1/2)=1/8 ft³, and each unit cube is 1/8 ft³, so 1 fits. This validates that packing matches the formula. No units on choices, but implicitly number of cubes. Mistake like thinking volume 1/8 means 8 small cubes, but here unit is defined as 1/2 ft, not 1 ft. Steps: divide each edge by unit edge, 1×1×1=1.

Question 12

A craft box is a rectangular prism that measures 0.6 m×0.5 m×0.4 m0.6\text{ m} \times 0.5\text{ m} \times 0.4\text{ m}. What is its volume?

  1. 0.15 m30.15\text{ m}^3
  2. 1.5 m31.5\text{ m}^3
  3. 0.12 m20.12\text{ m}^2
  4. 0.12 m30.12\text{ m}^3 (correct answer)
Explanation: To calculate the volume of a rectangular prism with fractional edge lengths, use the formula V = l × w × h, where you multiply the three dimensions, or V = base area × height, and understand that packing the prism with fractional unit cubes yields the same result. For example, with dimensions 2 ft × 3 ft × (1/2) ft, V = 2 × 3 × 0.5 = 3 ft³; alternatively, using base area, if the base is 5 m × 4 m = 20 m² and height 2.5 m, then V = 20 × 2.5 = 50 m³; packing with 1/2 ft cubes holds 24 cubes of 1/8 ft³ for 3 ft³, units cubic like m³. Another example: 2 ft × 3 ft × (1/2) ft = 3 ft³; 2.5 cm × 4 cm × 3 cm = 30 cm³; base 6 × 5 = 30 cm² × 2.5 cm = 75 cm³. For this craft box 0.6 m × 0.5 m × 0.4 m, V = 0.6 × 0.5 × 0.4 = 0.3 × 0.4 = 0.12 m³. Common errors include adding 0.6+0.5+0.4=1.5, decimal mistakes like 0.6×0.5=0.3 then 0.3×0.4=1.2 or 0.15, using m², sum formula, two dimensions only like 0.6×0.5=0.3 forgetting 0.4. Calculate by multiplying step-by-step 0.6×0.5=0.3, 0.3×0.4=0.12 m³; decimals like fractions. Base-height: base ×h; packing verifies; real craft storage; avoid adding, decimal errors, no cubic, missing dimension.

Question 13

A rectangular prism is filled by packing 12 ft×12 ft×12 ft\tfrac{1}{2}\text{ ft} \times \tfrac{1}{2}\text{ ft} \times \tfrac{1}{2}\text{ ft} unit cubes. The prism's dimensions are 2 ft×1 ft×12 ft2\text{ ft} \times 1\text{ ft} \times \tfrac{1}{2}\text{ ft}. How many of these 12\tfrac{1}{2}-foot cubes fit exactly in the prism?

  1. 4
  2. 6
  3. 8 (correct answer)
  4. 2
Explanation: This question tests calculating the volume of rectangular prisms with fractional edge lengths using V=lwh (multiply three edges) or V=bh (base area times height), understanding packing with fractional unit cubes gives same result. Volume V=lwh: multiply length, width, height (2 ft × 3 ft × (1/2) ft = 2×3×0.5=3 ft³, fractional edges multiply like any numbers). Or V=bh: base area l×w times height (base 5×4=20 m², height 2.5 m, volume 20×2.5=50 m³). Packing: fill prism with unit cubes (if edge 1/2 ft, unit cube is (1/2)³=1/8 ft³, box 2×1×(1/2) holds 2÷(1/2)×1÷(1/2)×(1/2)÷(1/2)=4×2×1=8 cubes of 1/8 ft³ each, total 8×(1/8)=1 ft³—same as V=lwh). Units: cubic (ft³, cm³, in³—length unit cubed). For this prism, number of cubes: (2/(1/2)) × (1/(1/2)) × ((1/2)/(1/2)) = 4×2×1=8. Common errors include multiplying dimensions instead of dividing or miscounting fits.

Question 14

Refer to the diagram. A company manufactures wooden blocks in two different sizes. Block A has dimensions shown, and Block B has each dimension exactly 12\frac{1}{2} the size of Block A's corresponding dimension. How many Block B pieces have the same total volume as 4 Block A pieces?

  1. 16 Block B pieces
  2. 24 Block B pieces
  3. 32 Block B pieces (correct answer)
  4. 48 Block B pieces
Explanation: When each dimension is scaled by 12\frac{1}{2}, the volume is scaled by (12)3=18\left(\frac{1}{2}\right)^3 = \frac{1}{8}. So Block B has 18\frac{1}{8} the volume of Block A. To equal the volume of 4 Block A pieces, we need 4÷18=4×8=324 ÷ \frac{1}{8} = 4 \times 8 = 32 Block B pieces. Choice A uses linear scaling. Choice B uses area scaling. Choice D incorrectly multiplies by 12.

Question 15

A rectangular prism has dimensions 34 ft×2 ft×112 ft\tfrac{3}{4}\text{ ft} \times 2\text{ ft} \times 1\tfrac{1}{2}\text{ ft}. What is its volume in cubic feet?

  1. 98 ft3\tfrac{9}{8}\text{ ft}^3
  2. 174 ft3\tfrac{17}{4}\text{ ft}^3
  3. 94 ft\tfrac{9}{4}\text{ ft}
  4. 94 ft3\tfrac{9}{4}\text{ ft}^3 (correct answer)
Explanation: This question tests calculating the volume of rectangular prisms with fractional edge lengths using V=lwh, where you multiply the three edges, and understanding that packing with fractional unit cubes yields the same result. Dimensions: 3/4 ft × 2 ft × 1 1/2 ft (convert to 3/2 ft), so V=(3/4)×2×(3/2)=(3/4)×(3/2)×2=(9/8)×2=18/8=9/4 ft³. Or base (3/4)×2=3/2 ft², times height 3/2 ft = (3/2)×(3/2)=9/4 ft³. Packing with 1/4 ft cubes: along 3/4 fits 3, 2 fits 8, 3/2 fits 6, total 3×8×6=144 cubes each (1/4)³=1/64 ft³, 144×1/64=144/64=9/4 ft³. Units ft³. Error like adding fractions wrongly to 3/4 + 2 + 3/2 = 17/4. Steps: convert mixed to improper; multiply (3/4)×2×(3/2)=9/4.

Question 16

A rectangular fish tank has a base area of 4124\frac{1}{2} square feet and a height of 1561\frac{5}{6} feet. If the tank is filled to 34\frac{3}{4} of its total capacity, how many cubic feet of water are in the tank?

  1. 63166\frac{3}{16} cubic feet (correct answer)
  2. 8148\frac{1}{4} cubic feet
  3. 1111 cubic feet
  4. 133413\frac{3}{4} cubic feet
Explanation: Convert to improper fractions: 412=924\frac{1}{2} = \frac{9}{2} and 156=1161\frac{5}{6} = \frac{11}{6}. Total volume = base area × height = 92×116=9912=334=814\frac{9}{2} \times \frac{11}{6} = \frac{99}{12} = \frac{33}{4} = 8\frac{1}{4} cubic feet. Water volume = 34×814=34×334=9916=6316\frac{3}{4} \times 8\frac{1}{4} = \frac{3}{4} \times \frac{33}{4} = \frac{99}{16} = 6\frac{3}{16} cubic feet. Choice B gives the total tank volume. Choice C incorrectly adds dimensions. Choice D uses wrong fraction operations.

Question 17

A rectangular prism is 212 cm2\tfrac{1}{2}\text{ cm} by 112 cm1\tfrac{1}{2}\text{ cm} by 2 cm2\text{ cm}. What is its volume in cubic centimeters?

  1. 15 cm315\text{ cm}^3
  2. 152 cm3\tfrac{15}{2}\text{ cm}^3 (correct answer)
  3. 92 cm3\tfrac{9}{2}\text{ cm}^3
  4. 6 cm36\text{ cm}^3
Explanation: To calculate the volume of a rectangular prism with fractional edge lengths, use the formula V = l × w × h, where you multiply the three dimensions, or V = base area × height, and understand that packing the prism with fractional unit cubes yields the same result. For example, with dimensions 2 ft × 3 ft × (1/2) ft, V = 2 × 3 × 0.5 = 3 ft³; alternatively, using base area, if the base is 5 m × 4 m = 20 m² and height 2.5 m, then V = 20 × 2.5 = 50 m³; packing with 1/2 ft cubes holds 24 × 1/8 = 3 ft³, units cubic like cm³. Another example: 2 ft × 3 ft × (1/2) ft = 3 ft³; 2.5 cm × 4 cm × 3 cm = 30 cm³; base 6 × 5 = 30 cm² × 2.5 cm = 75 cm³. For this prism 2(1/2) cm=5/2 cm, 1(1/2) cm=3/2 cm, 2 cm, V=(5/2)×(3/2)×2=(15/4)×2=30/4=15/2 cm³. Common errors include adding 2.5+1.5+2=6, wrong multiplication like (5/2)×(3/2)=15/4 then ×2=15/2 but maybe to 9/2 or 15, or 2.5×1.5=3.75×2=7.5 but fraction 15/2 correct. Convert mixed to improper, multiply numerators 5×3×2=30, denominators 2×2×1=4, 30/4=15/2 cm³. Base-height: choose base ×h; packing verifies; real small items; avoid adding, fraction errors, no cubic, missing dimension.

Question 18

A gift box is a rectangular prism with dimensions 12 ft×34 ft×4 ft\tfrac{1}{2}\text{ ft} \times \tfrac{3}{4}\text{ ft} \times 4\text{ ft}. What is its volume in cubic feet?

  1. 194 ft3\tfrac{19}{4}\text{ ft}^3
  2. 72 ft3\tfrac{7}{2}\text{ ft}^3
  3. 174 ft3\tfrac{17}{4}\text{ ft}^3
  4. 32 ft3\tfrac{3}{2}\text{ ft}^3 (correct answer)
Explanation: To calculate the volume of a rectangular prism with fractional edge lengths, use the formula V=l×w×hV = l \times w \times h, where you multiply the three dimensions, or V=base area×heightV = \text{base area} \times \text{height}, and understand that packing the prism with fractional unit cubes yields the same result. For example, with dimensions 2 ft×3 ft×12 ft2 \text{ ft} \times 3 \text{ ft} \times \frac{1}{2} \text{ ft}, V=2×3×0.5=3 ft3V = 2 \times 3 \times 0.5 = 3 \text{ ft}^3; alternatively, using base area, if the base is 5 m×4 m=20 m25 \text{ m} \times 4 \text{ m} = 20 \text{ m}^2 and height 2.5 m2.5 \text{ m}, then V=20×2.5=50 m3V = 20 \times 2.5 = 50 \text{ m}^3; packing with 12 ft\frac{1}{2} \text{ ft} cubes in a 2×3×122 \times 3 \times \frac{1}{2} prism holds 4×6×1=244 \times 6 \times 1 = 24 cubes, each 18 ft3\frac{1}{8} \text{ ft}^3, totaling 3 ft33 \text{ ft}^3, with units always in cubic form like ft3\text{ft}^3. Another example: a prism 2 ft×3 ft×12 ft2 \text{ ft} \times 3 \text{ ft} \times \frac{1}{2} \text{ ft} has V=2×3×12=3 ft3V = 2 \times 3 \times \frac{1}{2} = 3 \text{ ft}^3; or 2.5 cm×4 cm×3 cm2.5 \text{ cm} \times 4 \text{ cm} \times 3 \text{ cm}: V=2.5×4×3=30 cm3V = 2.5 \times 4 \times 3 = 30 \text{ cm}^3; or base 6×5=30 cm26 \times 5 = 30 \text{ cm}^2, height 2.5 cm2.5 \text{ cm}, V=30×2.5=75 cm3V = 30 \times 2.5 = 75 \text{ cm}^3. For this gift box with 12 ft×34 ft×4 ft\frac{1}{2} \text{ ft} \times \frac{3}{4} \text{ ft} \times 4 \text{ ft}, V=12×34×4=38×4=128=32 ft3V = \frac{1}{2} \times \frac{3}{4} \times 4 = \frac{3}{8} \times 4 = \frac{12}{8} = \frac{3}{2} \text{ ft}^3. Common errors include wrong fraction multiplication like 12+34+4=4.75\frac{1}{2} + \frac{3}{4} + 4 = 4.75 or 19/419/4, adding numerators, using sum, multiplying two only, or simplifying wrong to 17/417/4 or 7/27/2. To calculate, identify fractions, multiply 12×34=38\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}, then ×4=128=32 ft3\times 4 = \frac{12}{8} = \frac{3}{2} \text{ ft}^3; no mixed here but convert if needed. Base-height: choose base, multiply; packing confirms; real gift boxes for space; avoid adding, fraction errors, no cubic, missing dimension, sum use.

Question 19

A rectangular prism can be packed exactly with cubes that each have edge length 0.5 cm0.5\text{ cm}. The prism's dimensions are 2 cm×1.5 cm×1 cm2\text{ cm} \times 1.5\text{ cm} \times 1\text{ cm}. How many of the 0.5 cm0.5\text{ cm} cubes fit inside the prism?

  1. 12
  2. 6
  3. 24 (correct answer)
  4. 3
Explanation: This question tests understanding packing rectangular prisms with fractional unit cubes, which validates volume V=lwh by giving the same result. Prism 2 cm × 1.5 cm × 1 cm, unit cubes 0.5 cm edge: along 2 cm fits 4, 1.5 cm fits 3, 1 cm fits 2, total 4×3×2=24 cubes. Volume-wise, V=2×1.5×1=3 cm³, each cube (0.5)³=0.125 cm³, 3÷0.125=24 cubes. This shows packing matches formula. No units on choices, but number of cubes. Mistake like multiplying dimensions without dividing, 2×1.5×1=3, or adding to 4.5÷0.5=9 (close to choices). Steps: divide each dimension by 0.5, get 4,3,2; multiply 4×3×2=24.

Question 20

A rectangular prism has a volume of 7127\frac{1}{2} cubic inches. If the length is 2122\frac{1}{2} inches and the width is 1151\frac{1}{5} inches, what is the height of the prism?

  1. 52\frac{5}{2} inches (correct answer)
  2. 2122\frac{1}{2} inches
  3. 3183\frac{1}{8} inches
  4. 1141\frac{1}{4} inches
Explanation: Convert to improper fractions: Volume = 152\frac{15}{2}, length = 52\frac{5}{2}, width = 65\frac{6}{5}. Using V = lwh: 152=52×65×h\frac{15}{2} = \frac{5}{2} \times \frac{6}{5} \times h. First find 52×65=3010=3\frac{5}{2} \times \frac{6}{5} = \frac{30}{10} = 3. So 152=3h\frac{15}{2} = 3h, which gives h=152÷3=152×13=156=52h = \frac{15}{2} \div 3 = \frac{15}{2} \times \frac{1}{3} = \frac{15}{6} = \frac{5}{2} inches. Choice B repeats the length value. Choice C uses incorrect division. Choice D results from calculation errors.