12) are required to complete the piece of work in 18 days. How many workers a) 12 workers can complete a piece of work in 20 days. How many workers are needed to complete the work in 16 days? c) A garrison of 200 men has provisions for 45 days. For how many men would the provisions last only for 30 davc?

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UpStudy AI · Accepted Answer

Let's solve each part of the problem step by step:

a) If 12 workers can complete a piece of work in 20 days, we can calculate the number of workers needed to complete the work in 16 days using the concept of work done.

Let's denote:
- Number of workers needed to complete the work in 16 days as $$ x $$
- Total work done as 1 (since it's a unit of work)

We can set up the equation based on the work done:
$$
\frac{12 \text{ workers} \times 20 \text{ days}}{x \text{ workers} \times 16 \text{ days}} = 1
$$

Solving this equation will give us the number of workers needed to complete the work in 16 days.

b) If a garrison of 200 men has provisions for 45 days, we can calculate how many men would the provisions last for only 30 days using the concept of work done.

Let's denote:
- Number of men needed for 30 days as $$ y $$
- Total work done as 1 (since it's a unit of work)

We can set up the equation based on the work done:
$$
\frac{200 \text{ men} \times 45 \text{ days}}{y \text{ men} \times 30 \text{ days}} = 1
$$

Solving this equation will give us the number of men needed for 30 days.

Let's calculate the number of workers needed to complete the work in 16 days and the number of men needed for 30 days.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\frac{12\times 20}{x\times 16}=1$$
- step1: Find the domain:
  $$\frac{12\times 20}{x\times 16}=1,x\neq 0$$
- step2: Reduce the fraction:
  $$\frac{15}{x}=1$$
- step3: Cross multiply:
  $$15=x$$
- step4: Swap the sides:
  $$x=15$$
- step5: Check if the solution is in the defined range:
  $$x=15,x\neq 0$$
- step6: Find the intersection:
  $$x=15$$
The number of workers needed to complete the work in 16 days is 15 workers.

Now, let's calculate the number of men needed for 30 days.
Solve the equation by following steps:
- step0: Solve for $$y$$:
  $$\frac{200\times 45}{y\times 30}=1$$
- step1: Find the domain:
  $$\frac{200\times 45}{y\times 30}=1,y\neq 0$$
- step2: Reduce the fraction:
  $$\frac{300}{y}=1$$
- step3: Cross multiply:
  $$300=y$$
- step4: Swap the sides:
  $$y=300$$
- step5: Check if the solution is in the defined range:
  $$y=300,y\neq 0$$
- step6: Find the intersection:
  $$y=300$$
The number of men needed for 30 days is 300 men.

Therefore, the answers to the questions are:
a) 15 workers are needed to complete the work in 16 days.
c) 300 men would need the provisions for only 30 days.
a) 15 workers are needed to complete the work in 16 days. c) 300 men would need the provisions for only 30 days.

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