Question

a biology class is recording surface temperature to participate in the global learning and observations to benefit the environment (globe) program. beckett and nika together can enter the data they collected into the globe database in 2 min. it would take nika 3 fewer minutes than it would take beckett to enter the data alone. how long would it take each of them, working alone, to enter the data?

Explanation:

Step1: Let Beckett's time be \(x\) minutes

Let the time taken by Beckett to enter the data alone be \(x\) minutes. Then the time taken by Nika to enter the data alone is \((x - 3)\) minutes.
The rate of work of Beckett is \(\frac{1}{x}\) (data per minute) and the rate of work of Nika is \(\frac{1}{x-3}\) (data per minute).
When they work together, their combined rate is \(\frac{1}{2}\) (data per minute).

Step2: Set up the equation

Using the formula for combined work rate \(\frac{1}{x}+\frac{1}{x - 3}=\frac{1}{2}\)
Multiply through by \(2x(x - 3)\) (the LCM of \(x\), \(x-3\) and \(2\)):

$$2(x-3)+2x=x(x - 3)$$

Expand:

$$2x-6 + 2x=x^{2}-3x$$

$$4x-6=x^{2}-3x$$

Rearrange to get a quadratic equation:

$$x^{2}-7x + 6=0$$

Step3: Solve the quadratic equation

Factor the quadratic equation \(x^{2}-7x + 6=(x - 6)(x - 1)=0\)
Set each factor equal to zero:
\(x-6=0\) gives \(x = 6\)
\(x - 1=0\) gives \(x=1\)
If \(x = 1\), then \(x-3=1 - 3=-2\) (time cannot be negative). So we discard \(x = 1\)

Step4: Find Nika's time

If \(x = 6\) (Beckett's time), then Nika's time \(x-3=6 - 3=3\)

Answer:

Beckett takes \(6\) minutes and Nika takes \(3\) minutes.