Question

heather can complete one problem in thirty minutes and when she started she had three problems done. joel can complete one problem in twenty minutes and when he started he had two problems done. when will they have completed between 30 and 50 problems? which graph shows the solution to this problem, if x stands for the number of hours that have passed?

Explanation:

Step1: Calculate problems completed per hour

Heather can complete 1 problem in 30 minutes, so she can complete 2 problems per hour. She started with 3 problems done. So the number of problems Heather has completed after \(x\) hours is \(H = 2x+3\). Joel can complete 1 problem in 20 minutes, so he can complete 3 problems per hour. He started with 2 problems done. So the number of problems Joel has completed after \(x\) hours is \(J=3x + 2\).

Step2: Find total problems completed

The total number of problems they have completed together after \(x\) hours is \(T=(2x + 3)+(3x+2)=5x + 5\).

Step3: Set up inequality

We want to find when \(30\leqslant5x + 5\leqslant50\). First, solve the left - hand side of the inequality \(30\leqslant5x+5\). Subtract 5 from both sides: \(30 - 5\leqslant5x\), so \(25\leqslant5x\). Divide both sides by 5, we get \(x\geqslant5\). Then, solve the right - hand side of the inequality \(5x+5\leqslant50\). Subtract 5 from both sides: \(5x\leqslant50 - 5=45\). Divide both sides by 5, we get \(x\leqslant9\).

Answer:

The values of \(x\) (number of hours) satisfy \(5\leqslant x\leqslant9\). This is represented by a number line with closed circles at 5 and 9 and a line segment between them. So the answer is B.