Question

The difference between two numbers is 28. The larger number is eight less than twice the smaller number. Find both numbers.

Answer 1

The equation 6X plus 5Y equals 303X plus 7Y equals 285 represent the money collected from selling gift baskets in a school fundraising event if X represents the cost for each snack, gift basket and Y represents the cost for each chocolate gift basket. What is the cost for each chocolate basket?

Answer 2

Let's call the larger number "L" and the smaller number "S".

From the first sentence, we know that:

L - S = 28

From the second sentence, we know that:

L = 2S - 8

Now we can substitute the second equation into the first equation:

(2S - 8) - S = 28

S - 8 = 28

S = 36

Now we can use that value to find L:

L = 2S - 8

L = 2(36) - 8

L = 64

Therefore, the two numbers are 36 and 64.

Answer 3

We can solve this problem using a system of linear equations.

6X + 5Y = 303

X represents the cost for each snack, gift basket, and Y represents the cost for each chocolate gift basket, so we'll call the cost of a chocolate gift basket "C":

3X + 7C = 285

Now we have two equations in two variables. We can solve for C by using substitution:

6X + 5Y = 303

6X + 42C = 1710 (multiply the second equation by 6 to get 6X)

5Y - 42C = -1407 (subtract the second equation from the first)

5Y = 303 - 42C (add 42C to both sides)

Y = (303 - 42C) / 5 (divide both sides by 5)

Now we can substitute this expression for Y into the second equation:

3X + 7C = 285

3X + 7C = 285

3X + 7[(303 - 42C) / 5] = 285

Multiply both sides by 5 to eliminate the denominator:

15X + 7(303 - 42C) = 1425

15X + 2121 - 294C = 1425

15X - 294C = -696

Now we have two equations:

5Y = 303 - 42C

15X - 294C = -696

We can solve either equation for one variable in terms of the other and substitute into the other equation:

5Y = 303 - 42C

Y = (303 - 42C) / 5

15X - 294C = -696

15X - 294C = -696

15X - 58(303 - 5Y) = -3480 (substitute for C in terms of Y)

15X - 17574 + 290Y = -3480

15X + 290Y = 15894

Now we have one equation in one variable:

15X + 290Y = 15894

We can solve for Y:

Y = (15894 - 15X) / 290

Since Y represents the cost of a chocolate gift basket, we want a positive value for Y. We can start by plugging in X = 0:

Y = (15894 - 15(0)) / 290

Y = 54.8

However, this gives us a negative value for Y. We can try plugging in larger values for X until we get a positive value for Y:

Y = (15894 - 15(10)) / 290

Y = 50.8

Y = (15894 - 15(20)) / 290

Y = 46.8

Y = (15894 - 15(30)) / 290

Y = 42.8

So the cost of a chocolate gift basket is $42.80.