2.4 Comparing Fractions
the fractions a/b and c/d are equal exactly when
a x d and cxb
what are good benchmarks to use when comparing fractions
1/2, 1/4, 3/4 and 1/3 and 1
when we compared 3/4 and 5/6 by giving them cd, 4 x 6, what did we see
3/4 =18/24 5/6 =20/24 all we needed to compare was 3 x6 and 5 x4 because those were the numerators when we gave 3/4 and 5/6 a cd by multiplying the two denominators all we really had to do was to multiply the numerators of each fraction by the denominator of the other fraction and compare the resutling numbers
compare 3/4 and 5/6 by the common denominator method
4 x6 = 24 to compare; same conclusion reached by using the common denominator 12
consider comparing fractions by using common numerators by using a number like
as we subdivide the pieces b/w 0 and 1 into more pieces, each piece becomes smaller, and fractions with the same numerator, but increasing numerators, become smaller
a/b is greater than c/d when
axd is greater than cxb
a/b is less than c/d exactly when
axd is less than cxb
why is comparing fractions more complicated
b/c every fraction is equal to infinitely many other fractions unlike whole numbers, we can't always tell just by comparing digits whether the fractions are equal or whether one is greater than the other
which is greater 4/7 or 3/7
both fractions are described in terms of like parts, sevenths, so we only to have to see which one has more parts
which fraction is greater, 5/8 or 5/9
both fractions represent 5 parts, but 5/8 is 8ths and 5/9 is 5 ninths if an object is divided into 8 equal parts, then each part is larger than if the object is divided into 9 equal parts, fewer parts making up the same whole means that each part has to be larger. so eights are bigger than ninths
which fraction is greater, 4/235 or 6/301
common numerator of 12 to compare them
how can we compare two fractions by using common denominators
compare numerators
given two numbers in base ten, how can we determine which one is greater
comparing their digits
how can we compare two fractions by converting them to decimals
convert both fractions to decimals (dividing the numerator by the denominator) and comparing the decimals
what are the four general methods for determining whether two fractions are equal, or if not, which is greater
converting to decimals using common denominators cross multiplication using common numerators
what if we want to compare two fractions that don't have the same denominator?
give the two fractions a common denominator. it can be any cd, doesn't have to be the least cd a cd that always works is obtained by multiplying the two denominators
compare 1/2 and 3/5
if we divide an object into 5 equal pieces, then 2 and 1/2 of them would make half the object, and three pices is more than 2 and a half pieces, since 4/9 is less than a half, and 3/5 is more than a half, 3/5 is the larger fraction
reason that 4/9 is less than 1/2
if we divide an object into 9 equal pieces, then it would take four and a half pieces to make half of the object, and 4 pieces is less than 4 and a 1/2 pieces
what is cross multiplying
method for checking whether two fractions are equal or if not, which is greater. it is just a way to check if the numerators are equal when the two fractions are given the cd obtained by multiplying th two original denominators
when two fractions have the same numerator
the fraction with the greatest denominator is less than the fraction with the smaller denominator
if we want to compare two fractions and the fractions don't already have a common denominator,
then we can give the fractions a common numerator to compare them
to compare a/b and c/d
we can compare a x d/bxd and cxb and dxb b/c the denominators d x b and bx d are equal, we need to compare only the numerators a x d and cxb
what are we really doing when we give two fractions common denominators in order to compare the sizes of the fractions
when we give fractions cd, we create like parts; then we only have to see how many of those parts each fraction is made of