Question

Question #d071b

Answer 1

They will each have 120 stamps at the end of 4 months.

Explanation:

So, based on this, you can write two equations:
`56+16x=y`
`72+12x=y`

We know that they both have to equal the same thing after the same amount of months, after all. So, we can eliminate one of the variables like this:
`-(56+16x=y)`
`color(white)(....)72+12x=y`

Leading to:
`-56-16x=`-`y`
`color(white)(. . )72+12x= y`

Then you just add down, and you're left with:
`16-4x=0`

From there, you solve for `x`, and you can find out that `x=4,` which is how many months it will take.

Plug 4 into one of the first equations, and you've found your `y.`
`56+16(4)=y`
`56+64=y`
`120=y`

You can do both to check.
`72+12(4)=y`
`72+48=y`
`120=y`

Answer 2

In 4 months, Kai and Xian will each have 120 stamps.

Explanation:

The situation of each collector can be written as a linear equation.

Xian
Xian gets 16 stamps per month for `x` months, plus a starting number of 56 stamps.

That is a linear equation
`y = mx + b`

`y` = Xian's total stamps
`m` = the number received per month
`x` = the number of months
`b` = the number of stamps to start

`y =  m  x +  b`
`y = 16 x  + 56`
~ ~ ~ ~ ~ ~ ~ ~ ~

Kai
Kai gets 12 stamps per month, plus a starting number of 72 stamps.

`y =  m x  +  b`
`y = 12  x  + 72`
~ ~ ~ ~ ~ ~ ~ ~

The problem wants to know how many months until they have an equal number of stamps, so set the equations equal to each other and solve for `x`, already defined as "the number of months."

`16 x + 56 = 12 x + 72`
solve for `x`, already defined as "the number of months when they will be equal."

1) Subtract `12x` from both sides to get all the `x` terms together
`4x + 56 = 72`

2) Subtract `56` from both sides to isolate the `4x` term
`4x = 16`

3) Divide both sides by 4 to isolate `x`, already defined as "the number of months"
`x = 4` `larr` answer for "the number of months until they own an equal number of stamps"
~ ~ ~ ~ ~ ~ ~ ~

To find how many stamps they will each own in 4 months, sub in 4 for `x` in the equations

Xian
Xian already has 56 stamps, and will receive 16 a month for 4 months.

Xian: `16 (4) + 56`
`color(white)(..........)64   + 56`
`color(white)(..............)120` stamps after 4 months

Kai
Kai already has 72 stamps and will receive 12 a month for 4 months.

Kai: `12 (4) +  72`
`color(white)(.......)48    +  72`
`color(white)(.............)120` stamps after 4 months

~ ~ ~ ~ ~ ~ ~

Answer:
In 4 months, Kai and Xian will each have 120 stamps.