Question

I need help with math

Answer 1

To answer this question, we have the following information, and we need to translate this information into algebraic equations to solve the problem:

1. They made kites, wood sculptures, and puzzles. We can represent them as:

• Kites ---> k

,

• Wood Sculptures ---> w

,

• Puzzles ---> p

2. They sold 28 items, then, we have to say that, in total, they sold:

• k + w + p = 28

That is, the sum of kites, sculptures, and puzzles sum 28 items.

3. The club sold twice as many kites as puzzles. We can represent it as follows:

• 2k = p

4. The club sold twice as many puzzles as wood sculptures. That is:

• 2p = w

Then, we have the following equations:

`k+w+p=28`
`2k=p`
`2p=w`

The key in this problem is to have an equation with one unknown variable on it. Let try with the unknown variable p:

We have that:

`2k=p\Rightarrow(2k)/(2)=(p)/(2)\Rightarrow k=(p)/(2)`

(We divide by 2 to both sides of the equation.)

And we can substitute this value in the first equation. Likewise, we have that w = 2p.

Then, we have that:

k = p/2, w = 2p, and substituting these equations in the first formula, we have:

`k+w+p=28`
`(p)/(2)+2p+p=28`

Now, we can sum each of the variables as follows (we need to sum the coefficients):

`(p)/(2)+3p=28\Rightarrow(p+2\cdot3p)/(2)=28\Rightarrow(p+6p)/(2)=28\Rightarrow(7p)/(2)=28`

Now, if we multiply this equation by 2/7 to both sides of it, we have:

`(2)/(7)\cdot(7)/(2)p=(2)/(7)\cdot28\Rightarrow p=(28\cdot2)/(7)\Rightarrow p=(56)/(7)\Rightarrow p=8`

Therefore, we have that p is equal to 8.

Now we can substitute this value in the next equation to obtain w:

`2\cdot p=w\Rightarrow w=2\cdot8\Rightarrow w=16`

And, we also have that 2k = p. Then, we have:

`2k=p\Rightarrow(2k)/(2)=(p)/(2)\Rightarrow k=(p)/(2)\Rightarrow k=(8)/(2)\Rightarrow k=4`

Therefore, we have that Keith's club sold:

• p = 8

,

• w = 16

,

• k = 4

That is, Keith's club sold 8 puzzles, 16 wood sculptures, and 4 kites.

We can check this information as follows:

1. Using the first equation:

`k+w+p=28\Rightarrow4+16+8=28\Rightarrow28=28`

Which is always TRUE.

2. And also, checking for the next ones:

`2k=p\Rightarrow2\cdot4=8\Rightarrow8=8`
`2p=w\Rightarrow2\cdot8=16\Rightarrow16=16`

For these two cases, the identities are always TRUE. Then, the answers are correct.