Can you do a trial and error sum of ratio and proportion for me.

Sure! A **trial and error** method in **ratios and proportions** typically involves testing different values to see if they satisfy the conditions of the problem.

Let’s take a trial and error approach to solve a ratio and proportion problem.

**Example Problem:**
Two numbers are in the ratio 3:5, and their sum is 80. What are the two numbers?

### Step 1: Express the numbers in terms of a common multiple
If two numbers are in the ratio 3:5, we can represent them as:
\[
\text{First number} = 3x \quad \text{and} \quad \text{Second number} = 5x
\]
where \(x\) is the common multiple.

### Step 2: Use the given sum
We are told the sum of the two numbers is 80, so:
\[
3x + 5x = 80
\]
\[
8x = 80
\]

### Step 3: Solve for \(x\)
\[
x = \frac{80}{8} = 10
\]

### Step 4: Calculate the two numbers
Now, substitute \(x = 10\) back into the expressions for the two numbers:
\[
\text{First number} = 3 \times 10 = 30
\]
\[
\text{Second number} = 5 \times 10 = 50
\]

Thus, the two numbers are **30 and 50**.

### Using Trial and Error
To confirm the answer using trial and error:
1. Start by trying random values for \(x\) and check if they satisfy both the ratio and the sum.
   - Try \(x = 5\):
     \[
     3x + 5x = 3(5) + 5(5) = 15 + 25 = 40 \quad (\text{Too low!})
     \]
   - Try \(x = 15\):
     \[
     3(15) + 5(15) = 45 + 75 = 120 \quad (\text{Too high!})
     \]
   - Finally, try \(x = 10\), which we already calculated gives the correct sum of 80.

Would you like to explore further details or have any questions on this?  

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Here are five related questions for further practice:
1. If two numbers are in the ratio 2:7 and their sum is 81, what are the two numbers?
2. The ratio of two numbers is 4:9, and their difference is 45. What are the two numbers?
3. The numbers 36 and 54 are in what ratio?
4. If the ratio of two numbers is 5:6 and their product is 300, find the numbers.
5. Two numbers are in the ratio 8:13. If their difference is 35, what are the numbers?

**Tip:** When working with ratios, always think of them as a comparison of parts that can be scaled by a common factor.

Learn how to solve a ratio and proportion problem using trial and error. This example explores two numbers in the ratio 3:5 with a sum of 80. A step-by-step solution guides students through algebraic manipulation and trial and error techniques. Suitable for students in Grades 6-8.

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