SAT Math - How to order fractions from least to greatest or from greatest to least

Which of the following fractions is between 0.2 and 0.3?

$\dpi{100} \small \frac{1}{4}$

$\dpi{100} \small \frac{1}{6}$

$\dpi{100} \small \frac{1}{3}$

$\dpi{100} \small \frac{1}{2}$

$\dpi{100} \small \frac{2}{3}$

Explanation

$\dpi{100} \small \frac{1}{4}=0.25$

$\dpi{100} \small 0.2<0.25<0.3$

$\dpi{100} \small \frac{1}{6}\approx 0.167<0.2$

The other three choices are larger than 0.3.

Which of the following fractions is less than $\frac{3}{8}$ ?

$\frac{1}{3}$

$\frac{6}{16}$

$\frac{4}{7}$

$\frac{1}{2}$

$\frac{5}{13}$

Explanation

The purpose of this question is to find understandable values of fractions, which must be done using a common scale. Since looking at the denominators in the answers shows that there are many different numbers, decimal values would be easier to use.

The decimal equivalencies are as follows:

$\frac{6}{16}=0.375$

$\frac{4}{7}=0.571$

$\frac{1}{2}=0.5$

$\frac{5}{13}=0.385$

$\frac{1}{3}=0.333$

Therefore, we know that $\frac{3}{8}$ is equal to and find that the fraction that is less than that is $\frac{1}{3}$ , which is equal to .

Which of the following fractions is the smallest?

1/3, 115/276, 112/350, 1050/3330, 0.75/2

$\frac{1050}{3330}$

$\frac{0.75}{2}$

$\frac{112}{350}$

$\frac{1}{3}$

$\frac{115}{276}$

Explanation

1/3, 115/276, 112/350, 1050/3330, 0.75/2

First you need to put the fractions in ascending order:

1050/3330, 112/350, 1/3, 0.75/2, 115/276

Then choose the fraction with the smallest value (1050/3330).

Which of the following five numbers is the greatest?

$\frac{10}{13}$

$\frac{5}{7}$

$\frac{3}{4}$

$\frac{13}{17}$

$\frac{8}{11}$

Explanation

For each fraction, divide the numerator by the denominator. This is shown below for each fraction to three decimal places:

Division

From the decimal representations, $\frac{10}{13}$ can be seen to be the greatest of the five choices.

Which fraction falls between ½ and 3/4?

4/5

3/6

5/8

6/8

1/3

Explanation

The easiest method is to put each of these fractions into a calculator, and then place them in order on a number line to see which value falls in between 1/2 = 0.5 and ¾ = 0.75. Without a calculator you must do long division to find the value of the numerator (top number) divided by the denominator (bottom number) for each fraction. You can actually eliminate answers b. and d. because they are equal to 1/2 and ¾ respectively. 4/5= 0.80, 3/6 = 0.5, 5/8= 0.625, 6/8 = 0.75.

If Ben is taller than Jaime, Mary is taller than Ben, and Chris is taller than Mary. Who is the second tallest?

Chris

Mary

Ben

Jaime

Not possible to tell

Explanation

Using math symbols to dictate height we find that Ben>Jaime, Mary>Ben, Chris>Mary. Putting these in order we have Chris>Mary, Mary>Ben, Ben>Jaime. This shows that Mary is the second tallest.

Order from least to greatest:

$\frac{4}{7} , \frac{5}{8}, \frac{7}{12}$

$\frac{4 }{7} < \frac{7}{12} <\frac{5}{8}$

$\frac{4 }{7} <\frac{5}{8} < \frac{7}{12}$

$\frac{5}{8} < \frac{4 }{7} < \frac{7}{12}$

$\frac{5}{8} < \frac{7}{12} < \frac{4 }{7}$

$\frac{7}{12} < \frac{5}{8} < \frac{4 }{7}$

Explanation

Express all three fractions in terms of their least common denominator - the least common multiple of denominators 7, 8, and 12, which is 168. Do this by multiplying the numerator and denominator in each fraction by whatever number yields a product of 168 in the denominator.

$\frac{4 }{7} = \frac{4 \cdot 24 }{7 \cdot 24 } = \frac{96}{168}$

$\frac{5}{8} = \frac{5 \cdot 21 }{8 \cdot 21 } = \frac{105}{168 }$

$\frac{7}{12} = \frac{7 \cdot 14 }{12 \cdot 14 } = \frac{98}{168}$

;

it follows that

$\frac{96}{168}< \frac{98}{168} < \frac{105}{168 }$ ,

$\frac{4 }{7} < \frac{7}{12} <\frac{5}{8}$

Order the following fractions in descending order.

1/2, 2/3, 2/5, 3/4, 4/7

2/5, 1/2, 4/7, 2/3, 3/4

3/4, 2/3, 4/7, 1/2, 2/5

2/5, 1/2, 3/4, 4/7, 2/3

2/3, 4/7, 3/4, 1/2, 2/5

1/2, 2/3, 2/5, 3/4, 4/7

Explanation

Method 1:

Find the common denominator (420) and convert each fraction to this denominator:

1/2 = 210/420

2/3 = 280/420

2/5 = 168/420

3/4 = 315/420

4/7 = 240/420

Now sort by numerator, largest to smallest.

Method 2:

Divide each fraction to obtain a decimal. Sort the decimals, largest to smallest.

Order from greatest to least:

$- \frac{3}{13}, - \frac{4}{17}, - \frac{5}{19}$

$- \frac{3}{13}> - \frac{4}{17}>- \frac{5}{19}$

$- \frac{3}{13}>- \frac{5}{19}> - \frac{4}{17}$

$- \frac{5}{19}>- \frac{3}{13}> - \frac{4}{17}$

$- \frac{5}{19}> - \frac{4}{17}>- \frac{3}{13}$

$- \frac{4}{17}>- \frac{5}{19}>- \frac{3}{13}$

Explanation

Rewrite the numbers $- \frac{3}{13}, - \frac{4}{17}, - \frac{5}{19}$ in terms of the least common denominator, which is $13 \times 17 \times 19 = 4,199$ :

$- \frac{3}{13} = - \frac{3 \times 17 \times 19 }{13 \times 17 \times 19 } = - \frac{969}{4,199} = \frac{-969}{4,199}$

$- \frac{4}{17} = - \frac{4 \times 13 \times 19}{17 \times 13 \times 19 } = -\frac{988}{4,199} = \frac{-988}{4,199}$

$- \frac{5}{19} = - \frac{5 \times 13 \times 17 }{19 \times 13 \times 17} = -\frac{1,105}{4,199}= \frac{-1,105}{4,199}$

Comparing numerators, we see that

It follows that

$\frac{-969}{4,199} >\frac{ -988}{4,199} >\frac{ -1,105}{4,199}$

and

$- \frac{3}{13}> - \frac{4}{17}>- \frac{5}{19}$ ,

making this the correct order.

Order the following from least to greatest: three-fifths, seven-eighths, 0.5, sixteen-eighteenths, 97%, 17-fifths.

, three-fifths, seven-eighths, sixteen-eighteenths, , -fifths

, three-fifths, sixteen-eighteenths, seven-eighths, , -fifths

, three-fifths, seven-eighths, sixteen-eighteenths, -fifths,

, , three-fifths, seven-eighths, sixteen-eighteenths, -fifths

three-fifths, , , seven-eighths, sixteen-eighteenths, -fifths

Explanation

We need to order: three fifths, seven eights, 0.5, sixteen eighteenths, 97%, seventeen fifths.

Let us re-write each as a fraction: $\frac{3}{5}$ , $\frac{7}{8}$ , $\frac{1}{2}$ , $\frac{16}{18}=\frac{8}{9}$ , $\frac{97}{100}$ , and $\frac{17}{5}$ . We can now easily order these: