Complete the ratio table.
Understand the Problem
The question is asking to complete a ratio table where certain values are provided and others are missing. The goal is to determine the values that maintain the ratios consistent across the table.
Answer
The missing values are 58 and 87.
Answer for screen readers
The completed ratio table is:
| First Column | Second Column |
|---|---|
| 29 | 8 |
| 58 | 16 |
| 87 | 24 |
| 145 | 40 |
Steps to Solve
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Identify the Ratios We have two columns with values that must maintain a consistent ratio. The first column has values 29, ?, 116, and 145. The second column has values 8, 16, 24, and 40.
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Calculate the Ratios First, we will calculate the ratio of the known values in each row. The ratio of the first two known values (for example, 29 and 8) is: $$ \frac{29}{8} $$
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Find the Missing Value for 16 Using the ratio from the first row, we can find the missing value for the second row. We set up the proportion: $$ \frac{29}{8} = \frac{x}{16} $$
Cross-multiplying gives us: $$ 29 \cdot 16 = 8 \cdot x $$ $$ 464 = 8x $$ Now, divide by 8: $$ x = \frac{464}{8} = 58 $$
- Find the Missing Value for 24 For the next row, we again use the original ratio: $$ \frac{29}{8} = \frac{y}{24} $$
Cross-multiplying gives us: $$ 29 \cdot 24 = 8y $$ $$ 696 = 8y $$ Now, divide by 8: $$ y = \frac{696}{8} = 87 $$
- Check for the Last Row (with 40) Finally, we use the last known value in the second column (40) to find the corresponding value from the first column (z): $$ \frac{29}{8} = \frac{z}{40} $$
Cross-multiplying: $$ 29 \cdot 40 = 8z $$ $$ 1160 = 8z $$ Now, divide by 8: $$ z = \frac{1160}{8} = 145 $$
The completed ratio table is:
| First Column | Second Column |
|---|---|
| 29 | 8 |
| 58 | 16 |
| 87 | 24 |
| 145 | 40 |
More Information
The values in the completed ratio table maintain the ratio of ( \frac{29}{8} ), ensuring consistent relationships between the two columns.
Tips
- Not cross-multiplying correctly when setting up proportions. Always double-check calculations.
- Forgetting that ratios must be simplified to their lowest terms. Ensure fractions are reduced when necessary.
