Question

1. jordan wrote the following description: three fewer than one fourth of x is 12. write an equation to represent the description.
2. at a graduation dinner, an equal number of guests were seated at each of 3 large tables, and 7 late - arriving guests were seated at a smaller table. there were 37 guests in all. if n represents the number of people seated at each of the large tables, what equation represents the situation?
3. last night, 4 friends went out to dinner at a restaurant. they split the bill evenly. each friend paid $12.75 for his or her meal and each left the same amount for a tip, t. the total dinner bill including the tip was $61. what equation could you use to describe the situation?
4. mia buys 4\frac{1}{5} pounds of plums. the total cost after using a coupon for 55\text{c} off her entire purchase was $3.23. if c represents the cost of the plums in dollars per pound, what equation could represent the situation?

for 12 and 13, use the equation shown at the right.
\frac{9 + 3}{6}=15

1. describe a situation that the equation could represent.
2. reasoning would the situation you wrote for problem 12 work if the denominator in the equation were doubled? explain why or why not.

Explanation:

Step 1: Problem 8: Define equation

$x - \frac{1}{4}x = 12$

Step 2: Problem 9: Define equation

$3n + 7 = 37$

Step 3: Problem 10: Define equation

$4(12.75 + t) = 61$

Step 4: Problem 11: Define equation

$4\frac{1}{5}c - 0.55 = 3.23$

Step 5: Problem 12: Describe scenario

Example: I have 3 extra items, split all items into 6 groups, each group has 15 items. Find total original items $g$.

Step 6: Problem 13: Analyze doubled denominator

If denominator is 12, equation becomes $\frac{g+3}{12}=15$, so $g+3=180$, $g=177$. This would represent splitting the same total items into twice as many groups, each still with 15 items. It works if the scenario is adjusted to splitting into 12 equal groups of 15, instead of 6.

Answer:

1. $x - \frac{1}{4}x = 12$
2. $3n + 7 = 37$
3. $4(12.75 + t) = 61$
4. $4\frac{1}{5}c - 0.55 = 3.23$
5. Example: A teacher has $g$ pencils, adds 3 more, then divides all pencils equally into 6 bins. Each bin has 15 pencils. Find $g$, the original number of pencils.
6. It would work if the scenario is adjusted to splitting the combined quantity ($g+3$) into 12 equal groups (instead of 6) where each group has 15. The equation would represent a situation where the total quantity is split into twice as many equal parts, each still with a value of 15.