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Name: [Insert Name]
Date: [Insert Date]

Homework:

Read carefully and solve each problem. Show all work. (Read, Draw, Write)

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1. A restaurant receives shipments as follows: onions every 14 days, salad greens every 6 days, and potatoes every 20 days. Today, the restaurant received shipments of all three items.

- A. The next time the restaurant will receive shipments of both onions and salad greens is in ____ days.

- B. The next time the restaurant will receive shipments of both onions and potatoes is in ____ days.

2. Jessica needs to determine how many packages of cookies and cupcakes to buy for her family reunion. Her mother has asked her to buy the same number of each type of item, but no more than 100 of each. At the bakery, cookies are sold in packages of 12 and cupcakes are sold in packages of 9.

- The greatest number of packages that Jessica can buy is ____ packages of cookies and ____ packages of cupcakes.

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Instructions: Use the dropdown menus to correctly complete the statements.

Answer :

We need to determine two things:

1. When the restaurant will next receive certain shipments together.

2. How many packages of cookies and cupcakes Jessica can purchase so that she gets an equal number, with no more than 100 of each.

Let’s go through each part step by step.

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Part A: Restaurant Shipments

The restaurant receives:
- Onions every [tex]$14$[/tex] days,
- Salad greens every [tex]$6$[/tex] days,
- Potatoes every [tex]$7$[/tex] days.

(a) To find the next day the restaurant receives both onions and salad greens, we must compute the least common multiple (LCM) of [tex]$14$[/tex] and [tex]$6$[/tex].

The formula for the LCM of two numbers [tex]$a$[/tex] and [tex]$b$[/tex] is given by:
[tex]$$
\mathrm{LCM}(a,b) = \frac{a \times b}{\gcd(a,b)}
$$[/tex]

For [tex]$14$[/tex] and [tex]$6$[/tex]:
- First, find the greatest common divisor (GCD). We have [tex]$\gcd(14,6)=2$[/tex].
- Now, compute the LCM:
[tex]$$
\mathrm{LCM}(14,6)=\frac{14\times6}{2}=\frac{84}{2}=42\text{ days.}
$$[/tex]

(b) Next, to find the day the restaurant receives both onions and potatoes, compute the LCM of [tex]$14$[/tex] and [tex]$7$[/tex]:
- The GCD of [tex]$14$[/tex] and [tex]$7$[/tex] is [tex]$\gcd(14,7)=7$[/tex] (since [tex]$7$[/tex] divides [tex]$14$[/tex]).
- Thus,
[tex]$$
\mathrm{LCM}(14,7)=\frac{14\times7}{7}=\frac{98}{7}=14\text{ days.}
$$[/tex]

So, the answers for Part A are:
- The restaurant receives both onions and salad greens together every [tex]$42$[/tex] days.
- The restaurant receives both onions and potatoes together every [tex]$14$[/tex] days.

────────────────────────
Part B: Jessica’s Cookie and Cupcake Packages

Jessica needs to buy the same total number of cookies as cupcakes, but she cannot buy more than [tex]$100$[/tex] of each. The cookies come in packages of [tex]$12$[/tex], and the cupcakes come in packages of [tex]$9$[/tex].

Let [tex]$N$[/tex] be the total number of cookies (and also cupcakes). Since the cookies come in packages of [tex]$12$[/tex] and the cupcakes come in packages of [tex]$9$[/tex], [tex]$N$[/tex] must be a multiple of both [tex]$12$[/tex] and [tex]$9$[/tex]. This means [tex]$N$[/tex] is a common multiple of [tex]$12$[/tex] and [tex]$9$[/tex].

The smallest common multiple (the LCM) of [tex]$12$[/tex] and [tex]$9$[/tex] is calculated by:
[tex]$$
\mathrm{LCM}(12,9) = \frac{12 \times 9}{\gcd(12,9)}.
$$[/tex]

The GCD of [tex]$12$[/tex] and [tex]$9$[/tex] is [tex]$3$[/tex], so:
[tex]$$
\mathrm{LCM}(12,9)=\frac{12\times9}{3}=\frac{108}{3}=36.
$$[/tex]

Thus, the possible totals [tex]$N$[/tex] are multiples of [tex]$36$[/tex]: [tex]$36, 72, 108, \dots$[/tex]. However, because Jessica cannot purchase more than [tex]$100$[/tex] of each item, [tex]$N$[/tex] must be no more than [tex]$100$[/tex]. The largest multiple of [tex]$36$[/tex] that is less than or equal to [tex]$100$[/tex] is [tex]$72$[/tex] (since [tex]$108$[/tex] exceeds [tex]$100$[/tex]).

Now, determine the number of packages:
- For cookies (packages of [tex]$12$[/tex]):
[tex]$$
\text{Number of cookie packages}=\frac{72}{12}=6.
$$[/tex]
- For cupcakes (packages of [tex]$9$[/tex]):
[tex]$$
\text{Number of cupcake packages}=\frac{72}{9}=8.
$$[/tex]

────────────────────────
Final Answers:

A.
- The next time the restaurant will receive shipments of both onions and salad greens is in [tex]$\boxed{42}$[/tex] days.
- The next time the restaurant will receive shipments of both onions and potatoes is in [tex]$\boxed{14}$[/tex] days.

B.
- Jessica can buy [tex]$\boxed{6}$[/tex] packages of cookies and [tex]$\boxed{8}$[/tex] packages of cupcakes.

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