Ms. Ervine has a basket of grapes.

Question

Ms. Ervine has a basket of grapes. If she gives 6 grapes to each of her students, she will have 2 grapes left. If she gives 7 grapes to each student, she will have to buy 11 more grapes. How many students doe Ms. Ervine have? How many grapes are in the basket? many grapes are in the basket?

Answer 1

let x = the number of students.

let y = the number of grapes total.

Since each student is getting 6 grapes and we have two left over, this word problem turns into the following equation:

6x + 2 = y  (6x + 2 because y is the TOTAL number of grapes but we didn't use all of them so we have to add the 2 left over to = the total y)

Now if we change it so every student gets 7 grapes we have:

 

7x = y + 11 ( it's y + 11 because she had to buy 11 more grapes, but y is only the total that she had in her basket to begin with so we have to account for the fact that we added 11 more to the pool)

 

Solving this series of equations by substitution or elimination yields: the right answer but I prefer substitution so:

 

y = 6x+2(equation 1)

Substitute 6x+2 into equation 2 for y

7x = (6x+2) + 11

7x = 6x + 13

7x - 6x = 13

x = 13

Plug 13 into y = 6x + 2

y = 6(13)+2

y = 78 + 2

y = 80

 

x = 13 and y = 80.

Answer 2

Sounds like somewhere between 5 and 10 students.

 

Let's use the trial and error method: pick a number of students and see if the answer is correct. If not, we will try another number of students instead and see if that one works....

 

Let's try 6 students. 6 x 6 = 36, +2 = 38.   That's 38 grapes. 6 x 7 = 42.... she is only short by 4 grapes. It's NOT 6 students.

 

Maybe 7 students. 7 x 6 = 42, +2 = 44 grapes. 7 x 7 = 49, she is only short by 5 grapes.

 

What about 10 students? 10 x 6 = 60, + 2 = 62 grapes. 10 x 7 = 70; she is short by 8 grapes.

 

I am seeing a pattern ! Three more students, should be three more grapes (11 grapes) short, I think, Let's check.

 

13 students. 13 x 6 = 78 + 2 = 80 grapes. 13 x 7 = 91. That is 11 more than she has.

 

SO Ms. Ervine has 13 students and 80 grapes.