Twenty percent of the number of whites exceeded the number of reds by 10. Also, three fifths of the number of blues was exactly equal to 3 times the number of reds. How man of each were there if ten percent of the sum of the reds and blues was 94 less than the number of whites

Let:
- \( W \) represent the number of whites,
- \( R \) represent the number of reds,
- \( B \) represent the number of blues.

We are given the following conditions:

1. **Twenty percent of the number of whites exceeded the number of reds by 10**:
   \[
   0.2W = R + 10
   \]
   (This can be rewritten as \( R = 0.2W - 10 \)).

2. **Three fifths of the number of blues is exactly equal to 3 times the number of reds**:
   \[
   \frac{3}{5}B = 3R
   \]
   (This can be rewritten as \( B = 5R \)).

3. **Ten percent of the sum of the number of reds and blues is 94 less than the number of whites**:
   \[
   0.1(R + B) = W - 94
   \]
   (This can be rewritten as \( R + B = 10W - 940 \)).

### Steps to solve:

Using the equations:
1. From the first equation, express \( R \) in terms of \( W \):
   \[
   R = 0.2W - 10
   \]

2. Substitute this into the second equation for \( B \):
   \[
   B = 5R = 5(0.2W - 10) = W - 50
   \]

3. Now substitute the expressions for \( R \) and \( B \) into the third equation:
   \[
   (0.2W - 10) + (W - 50) = 10W - 940
   \]

   Simplifying the left-hand side:
   \[
   0.2W - 10 + W - 50 = 10W - 940
   \]
   \[
   1.2W - 60 = 10W - 940
   \]

4. Solve for \( W \):
   \[
   10W - 1.2W = 940 - 60
   \]
   \[
   8.8W = 880
   \]
   \[
   W = \frac{880}{8.8} = 100
   \]

5. Now substitute \( W = 100 \) back into the expressions for \( R \) and \( B \):
   \[
   R = 0.2(100) - 10 = 20 - 10 = 10
   \]
   \[
   B = 100 - 50 = 50
   \]

### Final Answer:
- The number of whites \( W = 100 \),
- The number of reds \( R = 10 \),
- The number of blues \( B = 50 \).

Would you like further details on any of the steps, or do you have any questions?

### Follow-up Questions:
1. How can we modify the equations if a different percentage for whites or reds was given?
2. How do we handle cases where fractions of one variable equal a sum involving others?
3. Can you generalize the solution method for problems involving percentage and ratios?
4. What other techniques could be used if decimal percentages were avoided?
5. How would this change if instead of a percentage, an absolute difference between colors was given?

### Tip:
Always break down word problems step by step by translating each condition into an equation before attempting to solve.

Twenty percent of the number of whites exceeded the number of reds by 10. Also, three fifths of the number of blues was exactly equal to 3 times the number of reds. How many of each were there if ten percent of the sum of the reds and blues was 94 less than the number of whites?

Learn how to solve a word problem involving percentages and ratios between whites, reds, and blues. This algebra problem involves setting up linear equations based on conditions about colors and solving for the number of each. Suitable for students in Grades 8-10.

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