The mean of 5 numbers is 34. Three of the numbers are 29, 26, and 35. If the remaining two numbers are in the ratio 1:3, find the numbers.

Answer

To solve the problem step by step, let's follow the logic used in the video transcript.

### Step 1: Understand the Mean
The mean of a set of numbers is calculated by dividing the total sum of the numbers by the count of the numbers. Here, we know the mean of 5 numbers is 34.

### Step 2: Calculate the Total Sum
Since the mean is 34 and there are 5 numbers, we can find the total sum of these numbers:
\[ \text{Total Sum} = \text{Mean} \times \text{Number of Observations} = 34 \times 5 = 170 \]

### Step 3: Sum of Known Numbers
We have three known numbers: 29, 26, and 35. Let's calculate their sum:
\[ 29 + 26 + 35 = 90 \]

### Step 4: Set Up the Equation for Unknowns
Let the remaining two numbers be \(x\) and \(3x\) (since they are in the ratio 1:3). The total sum of all five numbers can be expressed as:
\[ 90 + x + 3x = 170 \]
This simplifies to:
\[ 90 + 4x = 170 \]

### Step 5: Solve for \(x\)
Now, we can isolate \(x\):
\[ 4x = 170 - 90 \]
\[ 4x = 80 \]
\[ x = \frac{80}{4} = 20 \]

### Step 6: Find the Two Unknown Numbers
Now that we have \(x\), we can find the two unknown numbers:
- The first number is \(x = 20\)
- The second number is \(3x = 3 \times 20 = 60\)

### Conclusion
The two remaining numbers are 20 and 60.

### Final Answer
Therefore, the two numbers are **20 and 60**.

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