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Write Equations

Circle all of the equations that could represent the given situation. Explain your choices in the space provided.

1. Six friends went out to eat and split the \$66 bill evenly. How much money, in dollars [tex]p[/tex], did each friend pay?

a. [tex]p - 66 = 6[/tex]
b. [tex]6p = 66[/tex]
c. [tex]6 + p = 66[/tex]
d. [tex]\frac{66}{p} = 6[/tex]
e. [tex]66 = 6p[/tex]
f. [tex]\frac{1}{6}p = 66[/tex]

Explain your choices:

Let each friend pay [tex]$p$[/tex] dollars. Since six friends split a [tex]$\$[/tex]66[tex]$ bill evenly, we have the equation
$[/tex][tex]$6p = 66.$[/tex][tex]$
Solving for $[/tex]p[tex]$, we divide both sides by 6:
$[/tex][tex]$p = \frac{66}{6} = 11.$[/tex][tex]$

Now, let’s examine each option:

1. Option a:
$[/tex][tex]$p - 66 = 6$[/tex][tex]$
Solving, we add 66 to both sides:
$[/tex][tex]$p = 6 + 66 = 72,$[/tex][tex]$
which does not match $[/tex]p = 11[tex]$.

2. Option b:
$[/tex][tex]$6p = 66$[/tex][tex]$
Dividing both sides by 6:
$[/tex][tex]$p = \frac{66}{6} = 11,$[/tex][tex]$
which is correct.

3. Option c:
$[/tex][tex]$6 + p = 66$[/tex][tex]$
Subtracting 6 from both sides:
$[/tex][tex]$p = 66 - 6 = 60,$[/tex][tex]$
which is incorrect.

4. Option d:
$[/tex][tex]$\frac{66}{p} = 6$[/tex][tex]$
Multiply both sides by $[/tex]p[tex]$:
$[/tex][tex]$66 = 6p,$[/tex][tex]$
then divide by 6:
$[/tex][tex]$p = \frac{66}{6} = 11,$[/tex][tex]$
which is correct.

5. Option e:
$[/tex][tex]$66 = 6p$[/tex][tex]$
Divide by 6:
$[/tex][tex]$p = \frac{66}{6} = 11,$[/tex][tex]$
which is correct.

6. Option f:
$[/tex][tex]$\frac{1}{6}p = 66$[/tex][tex]$
Multiply both sides by 6:
$[/tex][tex]$p = 66 \times 6 = 396,$[/tex]$
which is incorrect.

Therefore, the equations that correctly represent the situation are those in options b, d, and e.