Answer :
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:
[tex]k+w+p=28[/tex][tex]2k=p[/tex][tex]2p=w[/tex]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:
[tex]2k=p\Rightarrow\frac{2k}{2}=\frac{p}{2}\Rightarrow k=\frac{p}{2}[/tex](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:
[tex]k+w+p=28[/tex][tex]\frac{p}{2}+2p+p=28[/tex]Now, we can sum each of the variables as follows (we need to sum the coefficients):
[tex]\frac{p}{2}+3p=28\Rightarrow\frac{p+2\cdot3p}{2}=28\Rightarrow\frac{p+6p}{2}=28\Rightarrow\frac{7p}{2}=28[/tex]Now, if we multiply this equation by 2/7 to both sides of it, we have:
[tex]\frac{2}{7}\cdot\frac{7}{2}p=\frac{2}{7}\cdot28\Rightarrow p=\frac{28\cdot2}{7}\Rightarrow p=\frac{56}{7}\Rightarrow p=8[/tex]Therefore, we have that p is equal to 8.
Now we can substitute this value in the next equation to obtain w:
[tex]2\cdot p=w\Rightarrow w=2\cdot8\Rightarrow w=16[/tex]And, we also have that 2k = p. Then, we have:
[tex]2k=p\Rightarrow\frac{2k}{2}=\frac{p}{2}\Rightarrow k=\frac{p}{2}\Rightarrow k=\frac{8}{2}\Rightarrow k=4[/tex]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:
[tex]k+w+p=28\Rightarrow4+16+8=28\Rightarrow28=28[/tex]Which is always TRUE.
2. And also, checking for the next ones:
[tex]2k=p\Rightarrow2\cdot4=8\Rightarrow8=8[/tex][tex]2p=w\Rightarrow2\cdot8=16\Rightarrow16=16[/tex]For these two cases, the identities are always TRUE. Then, the answers are correct.
