Select all ratios equivalent to 12:6.

Understand the Problem

The question asks to identify which ratios among the given options are equivalent to the ratio 12:6. This involves simplifying the ratio or comparing it with the provided options to determine equivalency.

Answer

The equivalent ratios are $ 18:9 $ and $ 10:5 $.

Steps to Solve

Simplify the Given Ratio First, simplify the ratio ( 12:6 ). This can be done by dividing both parts of the ratio by their greatest common divisor (GCD). The GCD of 12 and 6 is 6.

So, $$ 12 \div 6 = 2 $$ and $$ 6 \div 6 = 1 $$ Thus, the simplified ratio is ( 2:1 ).

Convert Each Option to the Same Format Next, convert each provided ratio to its simplest form to see if it matches ( 2:1 ).

For ( 18:9 ):
- GCD is 9,
- Simplified: ( 18 \div 9 : 9 \div 9 = 2:1 )
For ( 27:54 ):
- GCD is 27,
- Simplified: ( 27 \div 27 : 54 \div 27 = 1:2 )
For ( 70:48 ):
- GCD is 2,
- Simplified: ( 70 \div 2 : 48 \div 2 = 35:24 )
For ( 10:5 ):
- GCD is 5,
- Simplified: ( 10 \div 5 : 5 \div 5 = 2:1 )

Comparison of Ratios Now compare the simplified forms:

( 18:9 ) simplifies to ( 2:1 ) (equivalent).
( 27:54 ) simplifies to ( 1:2 ) (not equivalent).
( 70:48 ) simplifies to ( 35:24 ) (not equivalent).
( 10:5 ) simplifies to ( 2:1 ) (equivalent).

The ratios that are equivalent to ( 12:6 ) are ( 18:9 ) and ( 10:5 ).

More Information

Identifying equivalent ratios involves simplifying each ratio to find a common ratio representation. Ratios that simplify to the same form are equivalent and can be used interchangeably in problems involving proportions.

Tips

Not simplifying the ratios: Failing to simplify before comparison can lead to incorrect conclusions.
Misidentifying the GCD: Incorrectly identifying the greatest common divisor can cause incorrect simplifications.

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