SOLUTION: A total of y matches are needed to fill 30 matchboxes with the same number of matches in each box. If each box has three matchstick less, there will be enough sticks for 32 boxes.

Click here to see ALL problems on test

Question 1206975: A total of y matches are needed to fill 30 matchboxes with the same number of matches in each box. If each box has three matchstick less, there will be enough sticks for 32 boxes. What is the value of y?
Found 6 solutions by josgarithmetic, ikleyn, Theo, greenestamps, Edwin McCravy, MathTherapy:
Answer by josgarithmetic(39613)   (Show Source):
You can put this solution on YOUR website!
This could be a constant rates problem.

Matches per box, rate, r
Boxes, b
Matches, m
Basic Rule,

RATE BOXES MATCHES y/30 30 y y/32 32 y Difference 3

The simple equation,

Answer by ikleyn(52751)   (Show Source):
You can put this solution on YOUR website!
.
A total of y matches are needed to fill 30 matchboxes with the same number of matches
in each box. If each box has three matchstick less, there will be enough sticks for 32 boxes.
What is the value of y?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In the first scenario, there are matches in each matchbox. In the second scenario, there are matches in each matchbox. The difference is 3 matches; so we can write this equation - = 3. At this point, the setup is complete. To solve this equation, multiply both sides by 30*32. You will get 32y - 30y = 3*30*32 2y = 3*30*32 y = 3*30*16 = 1440. ANSWER. The value of y is 1440 .

Solved.

Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website!
let y = total number of matches.
let x = number of matches in each box.

your equation is y = 30 * x.
30 is the number of boxes.

if you take 3 matches away from each box, then you fill y matches with 32 boxes instead of 30.

that equation becomes y = 32 * (x - 3).

you have two equations that need to be solved simultaneously.

they are:

y = 30 * x
y = 32 * (x - 3)

since they are both equal to y, you can set them equal to each other to get:
30 * x = 32 * (x-3)
simplify to get 30 * x = 32 * x - 96
subtract 30 * x from both sides of the equation and add 96 to both sides of the equation to get 96 = 2 * x.
solve for x to get x = 48.

your first equation becomes y = 30 * 48 which is equal to 1440.
your second equation becomes y = 32 * 45 which is equal to 1440.

your solution is that the value of y is equal to 1440.

Answer by greenestamps(13195)   (Show Source):
You can put this solution on YOUR website!

You have three responses that all show variations of a standard formal algebraic method for solving the problem.

Here is a less formal method that can get you to the answer with less work. If your mental math is good, you can solve the whole problem mentally.

The number of matches is a constant, so the ratio of the number of matches in the two boxes is the same as the ratio of the numbers of boxes.

The ratio of the numbers of matchboxes is 30:32 or 15:16.

The ratio of the numbers of matches in each box is x-3:x.

Treating the ratios as fractions, the fractions 15/16 and (x-3)/x have to be equivalent.

In the fraction 15/16, the numerator and denominator differ by 1; in the second fraction, they differ by 3. So we can change 15/16 to a fraction equivalent to (x-3)/x by multiplying numerator and denominator by 3. That gives us the fraction 45/48.

That tells us that the total of y matches can be either 30 matchboxes with 48 matches each, or 32 matchboxes with 45 matches each. The total number of matches is then 48*30 = 1440 or 45*32 = 1440.

ANSWER: 1440

This same informal method can be made more formal; it will still be different than the method used by the other tutors.

Let x be the number of matches in each box if there are enough for 30 matchboxes.
Then x-3 is the number in each box if there are 32 matchboxes.
The number of matches is the same in either case:

So (again) there are x=48 matches in each box if there are 30 matchboxes and x-3=45 in each box if there are 32 matchboxes.

And of course we again get that the number of matches y is 48*30 = 45*32 = 1440.

Answer by Edwin McCravy(20054)   (Show Source):
You can put this solution on YOUR website!

y must be a common multiple of 30 and 32. The least common multiple of 30 and 32 is 480. So y is a multiple of 480, say y=480n, where n is a positive integer When you divide y=480n by 30 you get 16n, the larger number per box. When you divide y=480n by 32 you get 15n, the smaller number per box. They must differ by 3, and 16n and 15n differ by n, So n = 3 and y=480n = 480(3) = 1440. Edwin

Answer by MathTherapy(10549)   (Show Source):
You can put this solution on YOUR website!

A total of y matches are needed to fill 30 matchboxes with the same number of matches in each box. If each box has three matchstick less, there will be enough sticks for 32 boxes. What is the value of y? With y matches being in 30 matchboxes, and each matchbox having the same number of matches, number of matches in each of the 30 matchboxes = Three (3) less matchsticks in each of the 32 matchboxes results in matches in each of the 32 matchboxes Equally fitting y matchsticks into 32 boxes gives us matchsticks in each of the 32 matchboxes We then get: 16y - 3(480) = 15y ----- Multiplying by LCD, 480 16y - 15y = 3(480) y (Number of matchsticks) =