Equivalent fractions name the same number. If you cut every quarter of a bar in half, the whole still covers the same length. There are now eight equal pieces and each formerly shaded quarter accounts for two shaded eighths. Three quarters and six eighths therefore occupy the same share of the bar.
Change the number of pieces without changing the value
Multiplying a numerator and denominator by the same nonzero factor preserves the quotient. For positive integers, dividing both by a common factor reverses the operation. Dividing only one number changes the value. Adding the same amount to both numbers generally changes it too. One half becomes two thirds if you add one to both, but two fourths if you multiply both by two.
Find the missing numerator when the denominator is 20
Suppose a fraction with denominator 20 equals three fifths. Twenty contains four groups of five, so multiply the numerator 3 by 4 as well. The missing numerator is 12. The fractions locate the same point on the number line because the second divides each fifth into four smaller pieces.
Cross-products check equality when the scale factor is not obvious
To compare 14 over 21 with 22 over 33, multiply the first numerator by the second denominator and the second numerator by the first denominator. Both products are 462. This works because multiplying the proposed equality by both nonzero denominators clears them. You can also reduce the two fractions. Both reduce to two thirds.
For 14 over 21 and 21 over 30 the cross-products differ, so the fractions are unequal even though both numerators and denominators have increased.
