Boxes of wafers

Wafers measuring 11 3/4 cm, 4 2/5 cm, and 1 2/3 cm are packed in rectangular boxes measuring 23 1/2 cm, 26 2/5 cm, and 6 2/3 cm. What part of the volume of the box does one wafer make up?

Correct answer:

r =  1/48

Step-by-step explanation:

$$\begin{gathered} a=11\frac{3}{4}=11+\frac{3}{4}=\frac{11\cdot 4+3}{4}=\frac{44+3}{4}=\frac{47}{4}=11\frac{3}{4}=11.75\text{cm} \\ b=4\frac{2}{5}=4+\frac{2}{5}=\frac{4\cdot 5+2}{5}=\frac{20+2}{5}=\frac{22}{5}=4\frac{2}{5}=4.4\text{cm} \\ c=1\frac{2}{3}=1+\frac{2}{3}=\frac{1\cdot 3+2}{3}=\frac{3+2}{3}=\frac{5}{3}=1\frac{2}{3}\doteq 1.6667\text{cm} \\ A=23\frac{1}{2}=23+\frac{1}{2}=\frac{23\cdot 2+1}{2}=\frac{46+1}{2}=\frac{47}{2}=23\frac{1}{2}=23.5\text{cm} \\ B=26\frac{2}{5}=26+\frac{2}{5}=\frac{26\cdot 5+2}{5}=\frac{130+2}{5}=\frac{132}{5}=26\frac{2}{5}=26.4\text{cm} \\ C=6\frac{2}{3}=6+\frac{2}{3}=\frac{6\cdot 3+2}{3}=\frac{18+2}{3}=\frac{20}{3}=6\frac{2}{3}\doteq 6.6667\text{cm} \\ {V}_1=a\cdot b\cdot c=\frac{47}{4}\cdot \frac{22}{5}\cdot \frac{5}{3}=\frac{517}{6}=86\frac{1}{6}\doteq 86.1667{\text{cm}}^3 \\ {V}_2=A\cdot B\cdot C=\frac{47}{2}\cdot \frac{132}{5}\cdot \frac{20}{3}=4136{\text{cm}}^3 \\ r=V_1/V_2=\frac{517}{6}/4136=\frac{517}{6\cdot 4136}=\frac{517}{24816}=\frac{1}{48}=0.0208 \end{gathered}$$

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