What must be added to the given numbers so that they are in proportion? a. 1, 3, 10, 18 b. 23, 34, 56, 78

Answer

To solve the problem of what must be added to the given numbers so that they are in proportion, we will tackle each part step by step.

### Part a: 1, 3, 10, 18

1. **Assume a variable**: Let \( x \) be the number that we need to add to each of the given numbers. Thus, the new numbers will be:
- \( 1 + x \)
- \( 3 + x \)
- \( 10 + x \)
- \( 18 + x \)

2. **Set up the proportion**: We know that for four numbers \( a, b, c, d \) to be in proportion, the following must hold true:
\[ \frac{a}{b} = \frac{c}{d} \]
In our case:
\[ \frac{1 + x}{3 + x} = \frac{10 + x}{18 + x} \]

3. **Cross-multiply**: This gives us:
\[ (1 + x)(18 + x) = (3 + x)(10 + x) \]

4. **Expand both sides**:
- Left side:
\[ 1 \cdot 18 + 1 \cdot x + 18 \cdot x + x^2 = 18 + 19x + x^2 \]
- Right side:
\[ 3 \cdot 10 + 3 \cdot x + 10 \cdot x + x^2 = 30 + 13x + x^2 \]

5. **Set the equation**:
\[ 18 + 19x + x^2 = 30 + 13x + x^2 \]

6. **Cancel \( x^2 \) from both sides**:
\[ 18 + 19x = 30 + 13x \]

7. **Rearrange the equation**:
\[ 19x - 13x = 30 - 18 \]
\[ 6x = 12 \]

8. **Solve for \( x \)**:
\[ x = 2 \]

9. **Conclusion**: We must add \( 2 \) to each of the numbers \( 1, 3, 10, 18 \) to make them proportional. The new numbers will be \( 3, 5, 12, 20 \).

### Part b: 23, 34, 56, 78

1. **Assume a variable**: Let \( x \) be the number that we need to add to each of the given numbers. Thus, the new numbers will be:
- \( 23 + x \)
- \( 34 + x \)
- \( 56 + x \)
- \( 78 + x \)

2. **Set up the proportion**:
\[ \frac{23 + x}{34 + x} = \frac{56 + x}{78 + x} \]

3. **Cross-multiply**:
\[ (23 + x)(78 + x) = (34 + x)(56 + x) \]

4. **Expand both sides**:
- Left side:
\[ 23 \cdot 78 + 23x + 78x + x^2 = 1794 + 101x + x^2 \]
- Right side:
\[ 34 \cdot 56 + 34x + 56x + x^2 = 1904 + 90x + x^2 \]

5. **Set the equation**:
\[ 1794 + 101x + x^2 = 1904 + 90x + x^2 \]

6. **Cancel \( x^2 \) from both sides**:
\[ 1794 + 101x = 1904 + 90x \]

7. **Rearrange the equation**:
\[ 101x - 90x = 1904 - 1794 \]
\[ 11x = 110 \]

8. **Solve for \( x \)**:
\[ x = 10 \]

9. **Conclusion**: We must add \( 10 \) to each of the numbers \( 23, 34, 56, 78 \) to make them proportional. The new numbers will be \( 33, 44, 66, 88 \).