Sally is buying beads and string to make a necklace. The string costs $1.50, a package of 10 decorative beads costs $0.50, and a package of 25 plain beads costs $0.75. She can spend only $7.00 and needs 150 beads. How many packages of each type of bead should she buy?

Let's break this problem down step by step.

### Known information:
- The cost of the string is $1.50.
- A package of 10 decorative beads costs $0.50.
- A package of 25 plain beads costs $0.75.
- Sally needs **150 beads** in total.
- Sally can spend a total of **$7.00**.

### Define Variables:
Let:
- $ x $ = number of packages of decorative beads (10 beads per package).
- $ y $ = number of packages of plain beads (25 beads per package).

### Step 1: Total beads equation
Sally needs a total of 150 beads. Since each package of decorative beads contains 10 beads and each package of plain beads contains 25 beads, we can write:
\[
10x + 25y = 150
\]
### Step 2: Total cost equation
Sally's total budget is $7.00. Since the string costs $1.50, the remaining amount available for beads is:
\[
7.00 - 1.50 = 5.50 \, \text{dollars}
\]
Thus, the total cost of the beads must be $5.50. The cost of $ x $ packages of decorative beads is $ 0.50x $, and the cost of $ y $ packages of plain beads is $ 0.75y $. This gives the cost equation:
\[
0.50x + 0.75y = 5.50
\]

### Step 3: Solve the system of equations

We now have the system of equations:
1. $ 10x + 25y = 150 $
2. $ 0.50x + 0.75y = 5.50 $

#### First, simplify the first equation:
Divide through by 5:
\[
2x + 5y = 30 \tag{1}
\]

#### Now simplify the second equation:
Multiply through by 100 to eliminate decimals:
\[
50x + 75y = 550 \tag{2}
\]
Now divide through by 25:
\[
2x + 3y = 22 \tag{3}
\]

### Step 4: Solve by subtraction
Now, subtract equation (3) from equation (1):
\[
(2x + 5y) - (2x + 3y) = 30 - 22
\]
This simplifies to:
\[
2y = 8
\]
\[
y = 4
\]

### Step 5: Find $ x $
Substitute $ y = 4 $ into equation (3):
\[
2x + 3(4) = 22
\]
\[
2x + 12 = 22
\]
\[
2x = 10
\]
\[
x = 5
\]

### Conclusion:
Sally should buy:
- 5 packages of decorative beads, and
- 4 packages of plain beads.

### Verification:
- Total number of beads: $ 10(5) + 25(4) = 50 + 100 = 150 $ beads.
- Total cost: $ 0.50(5) + 0.75(4) = 2.50 + 3.00 = 5.50 $ dollars, which fits her budget.

Would you like further details or have any questions?

Here are 5 related questions:
1. What if Sally needed 200 beads instead of 150?
2. What would happen if the cost of the string increased to $2.00?
3. How would the problem change if the price of plain beads increased?
4. What if Sally had only $6.50 to spend instead of $7.00?
5. Can Sally afford one more package of decorative beads?

**Tip:** When solving systems of equations, elimination and substitution are both effective methods depending on how the equations are structured.

Learn how to solve a system of equations to help Sally buy beads for her necklace. With a total budget of $7.00, find out how many packages of decorative and plain beads she can buy to get exactly 150 beads. This step-by-step solution covers algebraic concepts such as systems of equations, elimination method, and cost constraints. Suitable for Grades 7-9 students.

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