2 : 7, 2 : 9, 7 : 2, 7 : 9

Understand the Problem

The question is presenting pairs of numbers in a ratio format. It seems to require identifying or interpreting relationships or comparisons among them, possibly in the context of ratios or proportions.

Answer

Ratios \(2:7\) and \(2:9\) can be compared as \( \frac{2}{7} > \frac{2}{9} \) and \(7:2\) and \(7:9\) can also be compared similarly.

Steps to Solve

Identify the Ratios The given ratios are:

(2 : 7)
(2 : 9)
(7 : 2)
(7 : 9)

Convert Ratios to Fractions Convert each ratio to its equivalent fraction:

( \frac{2}{7} )
( \frac{2}{9} )
( \frac{7}{2} )
( \frac{7}{9} )

Compare Each Fraction Compare the fractions to find which ones are equivalent or relate to one another.

To compare ( \frac{2}{7} ) and ( \frac{2}{9} ), find a common denominator:

The common denominator for (7) and (9) is (63).
( \frac{2}{7} = \frac{2 \times 9}{7 \times 9} = \frac{18}{63} )
( \frac{2}{9} = \frac{2 \times 7}{9 \times 7} = \frac{14}{63} )

Analyze Relationships Similar calculations can be done for the other ratios:

For ( \frac{7}{2} ) and ( \frac{7}{9} ):
The common denominator is (18).
( \frac{7}{2} = \frac{7 \times 9}{2 \times 9} = \frac{63}{18} )
( \frac{7}{9} = \frac{7 \times 2}{9 \times 2} = \frac{14}{18} )

Notice that:

( \frac{2}{7} > \frac{2}{9} )
( \frac{7}{2} > \frac{7}{9} )

Determine Proportionality Identify if any of the ratios can be derived from others by multiplying or dividing terms.

The relationships can be summarized as:

( \frac{2}{7} > \frac{2}{9} )
( \frac{7}{2} > \frac{7}{9} )

More Information

Ratios are useful for comparing quantities. They can represent rates, proportions, or simple relationships between two numbers. Understanding how to manipulate and compare these fractions helps in solving real-world problems in finance, science, and statistics.

Tips

Forgetting to simplify ratios to their lowest terms.
Miscalculating equivalent fractions due to incorrect common denominators or arithmetic mistakes.
Assuming that all ratios are proportionate without performing calculations.

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