Juliana says that she can use the patterns of equivalent ratios in the multiplication table below to write an infinite number of ratios that are equivalent to 6:10.

Question

A multiplication table.

Which statement explains whether Juliana is correct?
She is correct because she can multiply 6 and 10 by any number to form an equivalent ratio.
She is correct because 6:10 can be written as 1:2 and there are an infinite number of ratios for 1:2.
She is not correct because the multiplication table does not include multiples of 10.
She is not correct because 6:10 is equivalent to 3:5 and there are only 9 ratios in the multiplication table that are equivalent to 3:5.

Answer 1

Juliana is correct because she can multiply 6 and 10 by any number to form an equivalent ratio.

To elaborate, for any positive integer \( n \), the ratio \( 6n : 10n \) gives an equivalent ratio to \( 6 : 10 \). For example, if \( n = 1 \), the equivalent ratio is \( 6 : 10 \); if \( n = 2 \), it becomes \( 12 : 20 \); if \( n = 3 \), it becomes \( 18 : 30 \), and so on. This process can be continued indefinitely, showing that there are indeed an infinite number of equivalent ratios.