Question

Wendy and Miranda can assemble a table in 45 minutes. Working alone, it takes Miranda 48 minutes longer than Wendy to put this table together. How long would it take Wendy to assemble the table by herself?

Answer 1

So we know that Miranda and Wendy can assemble a table in 45 minutes. If W is the time it takes Wendy to assemble it alone then the time it takes Miranda to do it (M) is:

`M=48+W`

Now we are going to define a new quantity, what fraction of a table does Wendy make in 45 minutes? This number is given by:

`(45)/(W)`

So in 45 minutes Wendy makes 45/W tables. Analogously Miranda makes 45/M tables in 45 minutes. We need the sum of these two quantities to be equal to one since we know that working togther they assemble 1 table in 45 minutes. Then we get another equation so we have a system of two equations:

`\begin{gathered} (45)/(W)+(45)/(M)=1 \\ M=48+W \end{gathered}`

Then we substitute equation 2 into equation 1:

`\begin{gathered} (45)/(W)+(45)/(M)=(45)/(W)+(45)/(48+W)=1 \\ (45)/(W)+(45)/(48+W)=1 \\ (45)/(W)+(45)/(48+W)=45\cdot((1)/(W)+(1)/(48+W))=1 \\ \text{If we add the fractions we get:} \\ 45\cdot((1)/(W)+(1)/(48+W))=45\cdot(W+48+W)/(W\cdot(W+48))=1 \end{gathered}`

Then we get:

`\begin{gathered} 45\cdot(W+48+W)/(W\cdot(W+48))=45\cdot(2W+48)/(W^2+48W)=1 \\ \text{If we multiply both sides by }W^2+48W\colon \\ 45\cdot(2W+48)/(W^2+48W)\cdot(W^2+48W)=1\cdot(W^2+48W) \\ 45\cdot(2W+48)=W^2+48W \\ 90W+2160=W^2+48W \\ 0=W^2+48W-90W-2160 \\ W^2-42W-2160=0 \end{gathered}`

This last equation has two solutions. One of them is the time it takes Wendy to assemble the table by herself. In order to find it we must use the solving equation:

`\begin{gathered} \text{For a quadratic function: }aW^2+bW+c \\ \text{ We get:} \\ W=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \end{gathered}`

In this case a=1, b=-42 and c=-2160.Then the time we are looking for is given by:

`\begin{gathered} W=\frac{42\pm\sqrt[]{42^2-4\cdot1\cdot(-2160)}}{2}=\frac{42\pm\sqrt[]{10404}}{2} \\ W=(42\pm102)/(2) \\ W=(42+102)/(2)=(144)/(2)=72 \\ W=(42-102)/(2)-(60)/(2)=-30 \end{gathered}`

We have two solutions. One is negative which means it has no sense. Then the true solution is 72 so it takes Wendy 72 minutes to assemble the table by herself.