There were a total of 664 red and green beads in a box. 20% of the red beads and 2/3 of the green beads were removed from the box. If there were 422 beads left in the box, how many red beads were there in the box at first?


Let \( r \) represent the initial number of red beads, and \( g \) represent the initial number of green beads. We have the following equations based on the problem statement:

1. The total number of beads equation:
   \[
   r + g = 664
   \]

2. When 20% of the red beads and \(\frac{2}{3}\) of the green beads are removed, the remaining number of beads is 422:
   - 20% of the red beads removed means \( 0.2r \) beads were removed, leaving \( 0.8r \) red beads.
   - \(\frac{2}{3}\) of the green beads removed means \( \frac{2}{3}g \) beads were removed, leaving \( \frac{1}{3}g \) green beads.
   
   The remaining beads equation is:
   \[
   0.8r + \frac{1}{3}g = 422
   \]

### Step 1: Solve the system of equations

We start with the two equations:
1. \( r + g = 664 \)
2. \( 0.8r + \frac{1}{3}g = 422 \)

Let's solve the first equation for \( g \):
\[
g = 664 - r
\]

Substituting \( g = 664 - r \) into the second equation:
\[
0.8r + \frac{1}{3}(664 - r) = 422
\]

Simplify this equation:
\[
0.8r + \frac{664}{3} - \frac{r}{3} = 422
\]

To eliminate the fraction, multiply every term by 3:
\[
3 \times 0.8r + 664 - r = 3 \times 422
\]

Simplify:
\[
2.4r - r + 664 = 1266
\]

Combine like terms:
\[
1.4r = 1266 - 664
\]

\[
1.4r = 602
\]

Now, solve for \( r \):
\[
r = \frac{602}{1.4}
\]

\[
r = 430
\]

### Step 2: Find \( g \)

Using the value of \( r \), substitute back into \( g = 664 - r \):
\[
g = 664 - 430
\]

\[
g = 234
\]

### Conclusion

There were **430 red beads** in the box at first.

---

Would you like more details on the solution, or do you have any questions? Here are some related questions:

1. How do you verify the solution to ensure it's correct?
2. What are some other methods to solve this problem?
3. How do you set up equations from word problems?
4. How would the solution change if different percentages were removed?
5. Can this problem be solved graphically?

**Tip:** When dealing with percentage problems, setting up equations correctly is crucial to find accurate solutions.

Learn how to calculate the initial number of red beads in a box using equations and percentages. This problem involves setting up and solving a system of equations and understanding percentages in a mathematical context. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation