1. **State the problem:** We have a table of equivalent ratios with some missing values. The table is: | ? | 14 | | 6 | 28 | | 8 | 35 | | 10 | ? | | 12 | 49 | We need to find the missing values in the first column (top value) and the second column (value corresponding to 10). 2. **Identify the relationship:** The handwritten note "x2" suggests the second column values are twice the first column values. 3. **Check the given pairs:** - For 6 and 28: $28 \div 6 = \frac{28}{6} = 4.666...$ which is not 2. - For 8 and 35: $35 \div 8 = 4.375$ which is not 2. - For 10 and ?: unknown. - For 12 and 49: $49 \div 12 = 4.0833...$ which is not 2. So the "x2" note might mean something else or be misplaced. 4. **Check if the ratios are equivalent:** Calculate the ratio of second column to first column for each known pair: - $\frac{14}{?}$ unknown - $\frac{28}{6} = \frac{14}{3}$ - $\frac{35}{8}$ - $\frac{?}{10}$ unknown - $\frac{49}{12}$ 5. **Try to find a common ratio:** Check if the ratios $\frac{28}{6}$, $\frac{35}{8}$, and $\frac{49}{12}$ are equivalent. Calculate decimals: - $\frac{28}{6} = 4.666...$ - $\frac{35}{8} = 4.375$ - $\frac{49}{12} = 4.0833...$ They are not equal, so the ratios are not equivalent. 6. **Try to find a pattern in the second column:** Look at the second column: 14, 28, 35, ?, 49 - 14 to 28 doubles (x2) - 28 to 35 increases by 7 - 35 to ? unknown - ? to 49 unknown 7. **Try to find a pattern in the first column:** 6, 8, 10, 12 - Increases by 2 each time 8. **Assuming the ratio is constant, find the ratio using known pairs:** Try $\frac{14}{6} = 2.333...$ Try $\frac{28}{8} = 3.5$ Try $\frac{35}{10} = 3.5$ Try $\frac{49}{12} = 4.0833...$ No constant ratio. 9. **Since the problem states "Fill in the Missing Values in the Equivalent Ratio", assume the ratio is $\frac{14}{6} = \frac{x}{8} = \frac{y}{10} = \frac{49}{12}$** 10. **Find the ratio $k$ such that $\frac{14}{6} = k$:** $$k = \frac{14}{6} = \frac{7}{3}$$ 11. **Find missing values using $k = \frac{7}{3}$:** - For 8: $x = 8 \times k = 8 \times \frac{7}{3} = \frac{56}{3} = 18.666...$ - For 10: $y = 10 \times k = 10 \times \frac{7}{3} = \frac{70}{3} = 23.333...$ - For top value (corresponding to 14): $z$ such that $\frac{z}{?} = k$ but missing first value is unknown. 12. **Check if 49 corresponds to 12 with ratio $k$:** $$12 \times k = 12 \times \frac{7}{3} = 28$$ But given is 49, so the ratio is not consistent. 13. **Conclusion:** The problem likely wants the missing values in the first column and second column assuming the ratio between columns is constant. 14. **Use the pairs with known values to find the ratio:** Given pairs: - (6,14) - (8,28) - (10,35) - (12,49) Check ratios: - $\frac{14}{6} = 2.333...$ - $\frac{28}{8} = 3.5$ - $\frac{35}{10} = 3.5$ - $\frac{49}{12} = 4.0833...$ Ratios are not equal, but 28/8 and 35/10 both equal 3.5. 15. **Assuming ratio is 3.5, find missing top value:** $$\text{top value} = \frac{14}{3.5} = 4$$ 16. **Check if 12 corresponds to 49 with ratio 3.5:** $$12 \times 3.5 = 42 \neq 49$$ So 49 is inconsistent. 17. **Final missing values:** - Top value in first column: 4 - Value corresponding to 10 in second column: 35 (given) **Answer:** Top value = 4 Value corresponding to 10 = 35 --- **Summary:** The missing top value in the first column is 4, assuming the ratio between columns is 3.5 for the middle values.