How do answer this math question?

Question

Vern and Crystal are aides for Ms. Speedi. Vern can grade quizzes at the rate of three answers per minute. Four minutes after Vern starts to grade quizzes, Crystal joins him. She can grade at the rate of five answers per minute. Vern and Crystal continue grading at these rates until they are finished. The diagram shown in the figure shows Vern's and Crystal's contributions to grading a set of quizzes. The whole segment represents the completed job.

a) Let x represent the number of minutes Vern spent grading quizzes. Write an expression using x for each of the following descriptions.

(1) the number of minutes Crystal will be grading quizzes.

(2) the number of answers Vern will check.

(3) the number of answers Crystal will check.

(4) the total number of answers that will be checked by both Vern and Crystal.

b) After five minutes, how many answers has Vern checked? How many answers has Crystal checked?

c) After 10 minutes, how many answers has Vern checked? How many answers has Crystal checked?

d) Write an equation that states that Vern and Crystal have checked a total of 36 answers. Solve the equation to find out how long Vern has been grading when 36 answers have been checked. (Hint: use the pattern in parts (b) and (c) to help you write the equation.)

e) How long will Crystal work if she and Vern check 220 answers

Answer 1

A.
1. x-4 (She will be grading quizzes for 4 minutes less than Vern, because she started four minutes after he did.)

2. 3x (He grades three questions a minute, times the number of minutes.)

3. 5(x-4) (She grades 5 questions a minute, but is grading for 4 minutes less than Vern.)

4. 3x +5(x-4) (The total number of questions Vern graded plus the total Crystal graded.

B. 15 questions (3 per minute times five minutes) for Vern. 5 for Crystal (because she didn't start grading until the last minute of the 5 minutes, and she can grade 5 per minute)

C. 30 for Vern (again 3 per minute times 10 minutes) 30 for Crystal (6 minutes times 5 per minute)

D. 36=3x+5(x-4)

36=3x +5x -20 (Use the distributive property to multiply 5 x x and 5 x -4)

56 = 8x (add 20 to both sides. add 3x +5x)

7=x (divide both sides by 8)

E. 220=3x+5x-20

240=8x

30=x (Do the same thing you did for letter D, but use the total number of questions as 220, instead of 36) 30 minutes is how long Vern worked. 26 minutes would be the time Crystal worked, because she worked 4 minutes less.