Select all ratios equivalent to 6:3.
Understand the Problem
The question is asking to identify which of the given ratios (54:27, 10:5, and 20:1) are equivalent to the ratio 6:3. This involves simplifying each ratio and comparing them to find equivalence with 6:3.
Answer
The ratios equivalent to $6:3$ are $54:27$ and $10:5$.
Answer for screen readers
The ratios equivalent to $6:3$ are $54:27$ and $10:5$.
Steps to Solve
- Simplify the original ratio 6:3
To determine the equivalent ratios, we first simplify the ratio $6:3$.
We divide both numbers by their greatest common divisor (GCD), which is 3: $$ \frac{6}{3} : \frac{3}{3} = 2:1 $$
- Simplify the first ratio 54:27
Next, simplify the ratio $54:27$.
Again, we find the GCD, which in this case is 27: $$ \frac{54}{27} : \frac{27}{27} = 2:1 $$
- Simplify the second ratio 10:5
We then simplify the ratio $10:5$.
The GCD here is 5: $$ \frac{10}{5} : \frac{5}{5} = 2:1 $$
- Simplify the third ratio 20:1
Finally, we simplify the ratio $20:1$.
Since the second term is already 1, the ratio is already in its simplest form: $$ 20:1 $$
- Compare the simplified ratios
Now, we compare the simplified ratios to see which ones are equivalent to $2:1$:
- $54:27$ simplifies to $2:1$ (equivalent)
- $10:5$ simplifies to $2:1$ (equivalent)
- $20:1$ does not simplify to $2:1$ (not equivalent)
The ratios equivalent to $6:3$ are $54:27$ and $10:5$.
More Information
Both $54:27$ and $10:5$ simplify to the same ratio of $2:1$, which is equivalent to $6:3$. In proportional reasoning, two ratios are equivalent if they can be reduced to the same simplest form.
Tips
- A common mistake is failing to simplify the ratios correctly. Always ensure that you divide by the greatest common divisor of both terms.
- Another mistake is misidentifying the ratios once simplified; ensure to compare them to the correct simplified form of the original ratio.
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