Volume and Measurement Conversion: practice solutions, Grade 5 – download the PDF

1 Count the cubes ★★★

1. One layer is 3 cubes by 2 cubes: \(3 \times 2 = 6\) cubes.
2. There are 2 layers.
3. \(6 \times 2 = 12\). The volume is 12 cubic units.

2 Square units or cubic units? ★★★

1. The floor is a flat surface, so you measure its area in square units (for example square feet).
2. Sand fills a solid space, so you use cubic units (cubic feet).
3. A rope has one dimension, so you use linear units (feet or meters).
4. Space inside a box is volume, so cubic units.

3 Apply the formula ★★★

\(V = l \times w \times h = 7 \times 3 \times 4\). First \(7 \times 3 = 21\), then \(21 \times 4 = 84\).

The volume is 84 cm³.

4 A cube-shaped box ★★★

1. \(V = 5 \times 5 \times 5 = 125\). The box has a volume of 125 in³.
2. \(V = 10 \times 10 \times 10 = 1{,}000\). The block has a volume of 1,000 cm³.

5 Customary lengths ★★★

1. Feet to inches, multiply: \(4 \times 12 = 48\), so 4 ft = 48 in.
2. Yards to feet, multiply: \(5 \times 3 = 15\), so 5 yd = 15 ft.
3. Inches to feet, divide: \(36 \div 12 = 3\), so 36 in = 3 ft.
4. \(2 \times 5{,}280 = 10{,}560\), so 2 mi = 10,560 ft.

6 Metric lengths ★★★

1. \(7 \times 100 = 700\), so 7 m = 700 cm.
2. \(3 \times 1{,}000 = 3{,}000\), so 3 km = 3,000 m.
3. \(60 \div 10 = 6\), so 60 mm = 6 cm.
4. \(4.5 \times 100 = 450\), so 4.5 m = 450 cm.

7 Read the line plot ★★★

1. Three × are above \(\dfrac{3}{4}\), so \(\dfrac{3}{4}\) in appears most often.
2. Three leaves at \(\dfrac{3}{4}\) in and one at 1 in are longer than \(\dfrac{1}{2}\) in: 4 leaves.
3. \(1 - \dfrac{1}{4} = \dfrac{3}{4}\). The difference is \(\dfrac{3}{4}\) in.

8 Find the missing height ★★★

\(96 = 8 \times 6 \times h\), so \(96 = 48 \times h\).

\(h = 96 \div 48 = 2\). Check: \(8 \times 6 \times 2 = 96\).

The height is 2 in.

9 The aquarium ★★★

1. \(V = 24 \times 10 \times 12 = 240 \times 12 = 2{,}880\). The volume is 2,880 in³.
2. Half the height is 6 in, so \(V = 24 \times 10 \times 6 = 1{,}440\). The water has a volume of 1,440 in³, which is half of 2,880.

10 Using the base area ★★★

1. \(B = 6 \times 3 = 18\) ft².
2. \(V = 18 \times 5 = 90\). The volume is 90 ft³.

11 Packing cubes ★★★

Along the length: \(12 \div 3 = 4\) cubes. Along the width: \(9 \div 3 = 3\) cubes. Along the height: \(6 \div 3 = 2\) layers.

One layer has \(4 \times 3 = 12\) cubes, so the box holds \(12 \times 2 = 24\) cubes.

Check with volumes: the box is \(12 \times 9 \times 6 = 648\) in³, each cube is \(3 \times 3 \times 3 = 27\) in³, and \(648 \div 27 = 24\). 24 cubes fit.

12 Two prisms joined ★★★

Shed: \(10 \times 8 \times 7 = 560\) ft³.

Bin: \(4 \times 8 \times 5 = 160\) ft³.

The two solids do not overlap, so we add: \(560 + 160 = 720\).

The total volume is 720 ft³.

13 Capacity and weight ★★★

1. \(3 \times 4 = 12\), so 3 gal = 12 qt.
2. 2 cups make 1 pint: \(40 \div 2 = 20\), so 40 c = 20 pt.
3. \(6 \times 16 = 96\), so 6 lb = 96 oz.
4. \(3 \times 2{,}000 = 6{,}000\), so 3 tons = 6,000 lb.

14 Metric mass and capacity ★★★

1. \(2.5 \times 1{,}000 = 2{,}500\), so 2.5 kg = 2,500 g.
2. \(8{,}000 \div 1{,}000 = 8\), so 8,000 mL = 8 L.
3. \(350 \div 100 = 3.5\), so 350 cm = 3.5 m.
4. \(0.75 \times 1{,}000 = 750\), so 0.75 L = 750 mL.

15 Which is bigger? ★★★

1. 2 ft 6 in = \(2 \times 12 + 6 = 30\) in. Since \(31 > 30\), 31 in is greater.
2. 1.2 km = 1,200 m. Since \(1{,}200 > 1{,}150\), 1.2 km is greater.
3. 3 lb = \(3 \times 16 = 48\) oz. Since \(50 > 48\), 50 oz is greater.

16 True or false? ★★★

1. True. For example \(3 \times 4 \times 5 = 60\) and \(6 \times 4 \times 5 = 120\), which is twice as much.
2. False. \(4 \times 4 \times 4 = 64\), so the volume is 64 cm³. (16 is the area of one face.)
3. False. 1 yd = 3 ft = \(3 \times 12 = 36\) in.
4. True. \(5{,}000 \div 1{,}000 = 5\).

17 An L-shaped solid ★★★

Adding. Tall part: \(3 \times 4 \times 6 = 72\) cm³. Low part: \(5 \times 4 \times 2 = 40\) cm³. Total: \(72 + 40 = 112\) cm³.

Subtracting. The full block is \(8 \times 4 \times 6 = 192\) cm³. The missing corner is 5 cm long, 4 cm wide and \(6 - 2 = 4\) cm high: \(5 \times 4 \times 4 = 80\) cm³. Then \(192 - 80 = 112\).

Both methods give a volume of 112 cm³.

18 Filling a tank ★★★

1. \(V = 20 \times 10 \times 15 = 3{,}000\) cm³.
2. 3,000 cm³ = 3,000 mL = \(3{,}000 \div 1{,}000 = 3\) L. The tank holds 3 L.
3. \(3 - 2.4 = 0.6\) L, and \(0.6 \times 1{,}000 = 600\). 600 mL are still needed.

19 Two missing measures ★★★

1. \(210 = 35 \times h\), so \(h = 210 \div 35 = 6\). The height is 6 cm.
2. \(B = l \times w\), so \(35 = l \times 7\) and \(l = 35 \div 7 = 5\). The length is 5 cm.

Check: \(5 \times 7 \times 6 = 210\).

20 Doubling dimensions ★★★

1. \(3 \times 4 \times 5 = 60\) in³.
2. New height 10 in: \(3 \times 4 \times 10 = 120\) in³. That is \(120 \div 60 = 2\) times as large.
3. New size 6 by 8 by 10: \(6 \times 8 \times 10 = 480\) in³. That is \(480 \div 60 = 8\) times as large, because each of the three dimensions doubled: \(2 \times 2 \times 2 = 8\).

21 Punch for the party ★★★

1. 1 gal = 4 qt = 8 pt = 16 c, so 3 gal = \(3 \times 16 = 48\) cups.
2. Each cup holds 2 half-cups, so \(48 \div \dfrac{1}{2} = 48 \times 2 = 96\). The cooler fills 96 servings.

22 Sharing sand equally ★★★

1. The bags hold \(\dfrac{1}{8}\), \(\dfrac{3}{8}\), \(\dfrac{3}{8}\), \(\dfrac{4}{8}\) and \(\dfrac{4}{8}\) lb. The total is \(\dfrac{1+3+3+4+4}{8} = \dfrac{15}{8}\) lb, or \(1\dfrac{7}{8}\) lb.
2. \(\dfrac{15}{8} \div 5 = \dfrac{15}{40} = \dfrac{3}{8}\). Each bag gets \(\dfrac{3}{8}\) lb.