6.2 Equivalent Ratios

Equivalent Ratios are ratios that express the same underlying relationship between quantities even though their individual terms differ, because each ratio in the set is obtained from the others by multiplying or dividing every term by the same nonzero factor.

The Meaning of Equivalence Between Ratios

Equivalent Ratio Meaning

Two ratios are equivalent when they represent the same proportional relationship between their quantities, regardless of the actual magnitude of the terms. The ratios 1 : 2 and 4 : 8 are equivalent because in both cases the first quantity is half the second, even though the individual numbers differ.

Constructing Equivalent Ratios

Equivalent Ratio Construction by Scaling

An equivalent ratio is constructed by multiplying every term of the original ratio by the same nonzero number. Scaling a : b by a factor k produces the equivalent ratio ka : kb, since both terms grow by the identical multiplicative amount and their relationship to each other is preserved.

Ratio Reduction by Common Factors

Equivalent ratios can also be found by dividing every term by a common factor shared by all terms. Dividing 12 : 18 by the common factor 6 produces the equivalent ratio 2 : 3.

Ratio Simplest Form

A ratio is in simplest form when its terms share no common factor other than 1, meaning no further reduction is possible. Finding the simplest form is done by dividing all terms by their greatest common divisor.

Equivalent Ratios Across Number Types

Equivalent Ratios with Whole Numbers

When both terms of a ratio are whole numbers, equivalence is found by multiplying or dividing both terms by the same whole number, producing a new pair of whole numbers that preserves the original proportional relationship.

Equivalent Ratios with Fractions

When a ratio contains fractional terms, an equivalent whole-number ratio can be found by multiplying every term by the least common denominator of the fractions involved, clearing the fractions while preserving the ratio.

Equivalent Ratios with Decimals

A ratio containing decimal terms can be converted to an equivalent whole-number ratio by multiplying every term by a power of ten large enough to remove the decimal points from all terms simultaneously.

Equivalent Signed Ratios

When ratio terms carry signs, equivalence is preserved as long as every term is multiplied or divided by the same signed factor, so that the relative sign relationship between the terms remains unchanged. Multiplying both terms of a ratio by a negative number preserves equivalence, since both terms flip sign together.

Verifying Whether Ratios Are Equivalent

Common Multiplier Verification

One way to verify equivalence is to check whether a single common multiplier connects each pair of corresponding terms: dividing the second ratio's first term by the first ratio's first term should give the same result as dividing the second ratio's second term by the first ratio's second term.

Cross-Product Equivalence Verification

The most general test for equivalence between two ratios written as fractions is the cross-product test: two ratios a : b and c : d are equivalent exactly when the product of the outer terms equals the product of the inner terms.

Non-Equivalent Ratio Identification

Two ratios are not equivalent when no single common factor connects all corresponding terms, or equivalently when the cross-product test fails. The ratios 2 : 3 and 3 : 5 are not equivalent, since 2 times 5 does not equal 3 times 3.