Chapter 24, Problem 26A

To determine

The amount of carbon needed to produce $2.60$ tons of carbon steel required in the production of each type of tool.

Answer to Problem 26A

  $59.8$ pounds of carbon is required to make twist drill.

  $39$ pounds of carbon is required to make wrench.

  $54.6$ pounds of carbon is required to make threading die.

  $65$ pounds of carbon is required to make ordinary file.

  $49.4$ pounds of carbon is required to make Machinist's hammer.

  $44.2$ pounds of carbon is required to make Chuck jaw.

Explanation of Solution

Given information:

Below table is given for amount of carbon needed for different tool. Amount of carbon steel is $2.60$ tons.

  

Calculation:

As per the given information amount of carbon steel is $2.60$ tons.

Need to find the required percentage of carbon for different tool.

  $1.$Twist drill :

As we know the proportion to find percentage is given as

  $\frac{P}{B}=\frac{R}{100}$   where, $P=\text{percentage, }B=\text{ base and }R=\text{rate}$

Here, amount of carbon steel will be base $B=$$2.60$ tons, percent of carbon in carbon steel is $1.15\%$ i.e. $R=$$1.15$.

Putting these value in standard proportion,

  $\begin{array}{l}\frac{P}{B}=\frac{R}{100}\\ \Rightarrow \frac{P}{2.60}=\frac{1.15}{100}\text{ {Put }B=2.60\text{ and }R=1.15\text{}}\\ \text{Applying cross multiplication,}\\ \Rightarrow 100P=1.15\times 2.60\\ \Rightarrow P=\frac{1.15\times 2.60}{100}\\ \Rightarrow P=0.0299\text{ tons}\\ \Rightarrow P=0.0299\times 2000\text{ {Since, 1 ton}=2000\text{ pound}}\\ \Rightarrow P=59.8\text{ pounds}\text{.}\end{array}$

Hence, $59.8$ pounds of carbon is required to make twist drill.

  $2.$Wrench :

As we know the proportion to find percentage is given as

  $\frac{P}{B}=\frac{R}{100}$   where, $P=\text{percentage, }B=\text{ base and }R=\text{rate}$

Here, amount of carbon steel will be base $B=$$2.60$ tons, percent of carbon in carbon steel is $0.75\%$ i.e. $R=$$0.75$.

Putting these value in standard proportion,

  $\begin{array}{l}\frac{P}{B}=\frac{R}{100}\\ \Rightarrow \frac{P}{2.60}=\frac{0.75}{100}\text{ {Put }B=2.60\text{ and }R=0.75\text{}}\\ \text{Applying cross multiplication,}\\ \Rightarrow 100P=0.75\times 2.60\\ \Rightarrow P=\frac{0.75\times 2.60}{100}\\ \Rightarrow P=0.0195\text{ tons}\\ \Rightarrow P=0.0195\times 2000\text{ {Since, 1 ton}=2000\text{ pound}}\\ \Rightarrow P=39\text{ pounds}\text{.}\end{array}$

Hence, $39$ pounds of carbon is required to make wrench.

  $3.$Threading die :

As we know the proportion to find percentage is given as

  $\frac{P}{B}=\frac{R}{100}$   where, $P=\text{percentage, }B=\text{ base and }R=\text{rate}$

Here, amount of carbon steel will be base $B=$$2.60$ tons, percent of carbon in carbon steel is $1.05\%$ i.e. $R=$$1.05$.

Putting these value in standard proportion,

  $\begin{array}{l}\frac{P}{B}=\frac{R}{100}\\ \Rightarrow \frac{P}{2.60}=\frac{1.05}{100}\text{ {Put }B=2.60\text{ and }R=1.05\text{}}\\ \text{Applying cross multiplication,}\\ \Rightarrow 100P=1.05\times 2.60\\ \Rightarrow P=\frac{1.05\times 2.60}{100}\\ \Rightarrow P=0.0273\text{ tons}\\ \Rightarrow P=0.0273\times 2000\text{ {Since, 1 ton}=2000\text{ pound}}\\ \Rightarrow P=54.6\text{ pounds}\text{.}\end{array}$

Hence, $54.6$ pounds of carbon is required to make threading die.

  $4.$Ordinary file :

As we know the proportion to find percentage is given as

  $\frac{P}{B}=\frac{R}{100}$   where, $P=\text{percentage, }B=\text{ base and }R=\text{rate}$

Here, amount of carbon steel will be base $B=$$2.60$ tons, percent of carbon in carbon steel is $1.25\%$ i.e. $R=$$1.25$.

Putting these value in standard proportion,

  $\begin{array}{l}\frac{P}{B}=\frac{R}{100}\\ \Rightarrow \frac{P}{2.60}=\frac{1.25}{100}\text{ {Put }B=2.60\text{ and }R=1.25\text{}}\\ \text{Applying cross multiplication,}\\ \Rightarrow 100P=1.25\times 2.60\\ \Rightarrow P=\frac{1.25\times 2.60}{100}\\ \Rightarrow P=0.0325\text{ tons}\\ \Rightarrow P=0.0325\times 2000\text{ {Since, 1 ton}=2000\text{ pound}}\\ \Rightarrow P=65\text{ pounds}\text{.}\end{array}$

Hence, $65$ pounds of carbon is required to make ordinary file.

  $5.$Machinist's hammer :

As we know the proportion to find percentage is given as

  $\frac{P}{B}=\frac{R}{100}$   where, $P=\text{percentage, }B=\text{ base and }R=\text{rate}$

Here, amount of carbon steel will be base $B=$$2.60$ tons, percent of carbon in carbon steel is $0.95\%$ i.e. $R=$$0.95$.

Putting these value in standard proportion,

  $\begin{array}{l}\frac{P}{B}=\frac{R}{100}\\ \Rightarrow \frac{P}{2.60}=\frac{0.95}{100}\text{ {Put }B=2.60\text{ and }R=0.95\text{}}\\ \text{Applying cross multiplication,}\\ \Rightarrow 100P=0.95\times 2.60\\ \Rightarrow P=\frac{0.95\times 2.60}{100}\\ \Rightarrow P=0.0247\text{ tons}\\ \Rightarrow P=0.0247\times 2000\text{ {Since, 1 ton}=2000\text{ pound}}\\ \Rightarrow P=49.4\text{ pounds}\text{.}\end{array}$

Hence, $49.4$ pounds of carbon is required to make Machinist's hammer.

  $6.$Chuck jaw :

As we know the proportion to find percentage is given as

  $\frac{P}{B}=\frac{R}{100}$   where, $P=\text{percentage, }B=\text{ base and }R=\text{rate}$

Here, amount of carbon steel will be base $B=$$2.60$ tons, percent of carbon in carbon steel is $0.85\%$ i.e. $R=$$0.85$.

Putting these value in standard proportion,

  $\begin{array}{l}\frac{P}{B}=\frac{R}{100}\\ \Rightarrow \frac{P}{2.60}=\frac{0.85}{100}\text{ {Put }B=2.60\text{ and }R=0.85\text{}}\\ \text{Applying cross multiplication,}\\ \Rightarrow 100P=0.85\times 2.60\\ \Rightarrow P=\frac{0.85\times 2.60}{100}\\ \Rightarrow P=0.0221\text{ tons}\\ \Rightarrow P=0.0221\times 2000\text{ {Since, 1 ton}=2000\text{ pound}}\\ \Rightarrow P=44.2\text{ pounds}\text{.}\end{array}$

Hence, $44.2$ pounds of carbon is required to make Chuck jaw.

Therefore,

  $59.8$ pounds of carbon is required to make twist drill.

  $39$ pounds of carbon is required to make wrench.

  $54.6$ pounds of carbon is required to make threading die.

  $65$ pounds of carbon is required to make ordinary file.

  $49.4$ pounds of carbon is required to make Machinist's hammer.

  $44.2$ pounds of carbon is required to make Chuck jaw.