Mastering Math: Comparing Numbers, Ordering Fractions, Decimals and Percents

Pair of Fractions	Common Denominator
$\frac{2}{3}$ and $\frac{5}{3}$	$3$
$\frac{8}{1 0}$ and $\frac{5}{1 0}$	$10$
$\frac{1 1}{1 7}$ and $\frac{6}{1 7}$	$17$

Pair of Fractions

Common Denominator

\frac{2}{3}

and

\frac{5}{3}

3

\frac{8}{1 0}

and

\frac{5}{1 0}

10

\frac{1 1}{1 7}

and

\frac{6}{1 7}

17

Fractions	Denominators	Multiples of Denominators	Common Denominators	LCM of Denominators (LCD)
$\frac{2}{3}$ and $\frac{1}{2}$	$3$ and $2$	$Multiples of 3 : Multiples of 2 : 3, 6, 9, 12, 15, \dots 2, 4, 6, 8, 10, 12, \dots$	$6,$ $12$	$6$
$\frac{5}{6}$ and $\frac{1}{4}$	$6$ and $4$	$Multiples of 6 : Multiples of 4 : 6, 12, 18, 24, 30, \dots 4, 8, 12, 16, 20, 24, \dots$	$12,$ $24$	$12$
$\frac{1}{4}$ and $\frac{5}{2}$	$4$ and $2$	$Multiples of 4 : Multiples of 2 : 4, 8, 12, \dots 2, 4, 6, 8, 10, 12, \dots$	$4,$ $8,$ $12$	$4$

Fractions

Denominators

Multiples of Denominators

Common Denominators

LCM of Denominators (LCD)

\frac{2}{3}

and

\frac{1}{2}

3

and

2

Multiples of 3 : Multiples of 2 : 3, 6, 9, 12, 15, \dots 2, 4, 6, 8, 10, 12, \dots

6,

12

6

\frac{5}{6}

and

\frac{1}{4}

6

and

4

Multiples of 6 : Multiples of 4 : 6, 12, 18, 24, 30, \dots 4, 8, 12, 16, 20, 24, \dots

12,

24

12

\frac{1}{4}

and

\frac{5}{2}

4

and

2

Multiples of 4 : Multiples of 2 : 4, 8, 12, \dots 2, 4, 6, 8, 10, 12, \dots

4,

8,

12

4

Fractions	$\frac{4}{1 1}$ and $\frac{9}{1 1}$
Numerators	$4 < 9$
Conclusion	$\frac{4}{1 1} < \frac{9}{1 1}$

Fractions

\frac{4}{1 1}

and

\frac{9}{1 1}

Numerators

4 < 9

Conclusion

\frac{4}{1 1} < \frac{9}{1 1}

Same Denominator	Same Numerator
The greater numerator, the greater the fraction.	The smaller denominator, the greater the fraction.

Same Denominator

Same Numerator

The greater numerator, the greater the fraction.

The smaller denominator, the greater the fraction.

Denominator	Calculating the Quotient	Factor
$6$	$\frac{6 0}{6}$	$10$
$20$	$\frac{6 0}{2 0}$	$3$
$3$	$\frac{6 0}{3}$	$20$
$5$	$\frac{6 0}{5}$	$12$

Denominator

Calculating the Quotient

Factor

6

\frac{6 0}{6}

10

20

\frac{6 0}{2 0}

3

3

\frac{6 0}{3}

20

5

\frac{6 0}{5}

12

Fraction	Equivalent Fraction
$\frac{5}{6}$	$\frac{5 0}{6 0}$
$\frac{5}{2 0}$	$\frac{1 5}{6 0}$
$\frac{2}{3}$	$\frac{4 0}{6 0}$
$\frac{1}{5}$	$\frac{1 2}{6 0}$

Fraction

Equivalent Fraction

\frac{5}{6}

\frac{5 0}{6 0}

\frac{5}{2 0}

\frac{1 5}{6 0}

\frac{2}{3}

\frac{4 0}{6 0}

\frac{1}{5}

\frac{1 2}{6 0}

Consider the numbers found by the students during their research about different flowers. One number is a fraction, one is a decimal number, one is a mixed number, and one is a percent.

\frac{4}{5}, 0.91, 1 \frac{2}{9}, 78 %

To compare the numbers, they need to be written in the same form. It does not matter which format is chosen, but all numbers will be rewritten as percents in this solution.

Fraction

Start by rewriting the fraction

\frac{4}{5}

as a percent. This can be done by multiplying the fraction by

100,

then dividing the new numerator by the denominator. Start with the multiplication.

\frac{4}{5}

Mult

Multiply by $100$

\frac{4}{5} \cdot 100

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

\frac{4 \cdot 1 0 0}{5}

Multiply

\frac{4 0 0}{5}

Next, divide the numerator by the denominator and add a percent sign.

\frac{4 0 0}{5} = 80 %

Therefore,

\frac{4}{5}

corresponds to

80 % .

Decimal Number

Now consider the decimal number

0.91 .

Rewrite it as a percent by multiplying it by

100

and adding a percent sign. Remember that a number can be easily multiplied by

100

by moving its decimal point two places to the right.

Finally, add a percent sign.

0.91 \cdot 100 = 91 %

Mixed Number

Consider the mixed number

1 \frac{2}{9} .

It consists of the integer

1

and the fraction part

\frac{2}{9} .

Begin by rewriting the fraction part as a decimal. Divide the numerator by the denominator to two decimal places by using long division.

The fraction corresponds to about

0.22 .

Next, join the integer part of

1

and the decimal part of

0.22

to write the final decimal form of

1 \frac{2}{9} .

1 \frac{2}{9} \approx 1.22

Finally, multiply the number by

100

and add a percent sign.

1.22 \cdot 100 = 122 %

Therefore,

1 \frac{2}{9}

corresponds to

122 % .

Comparison

Gather all the numbers and their corresponding percent forms.

\frac{4}{5} ↓ 80 % 0.91 ↓ 91 % 1 \frac{2}{9} ↓ 122 % 78 % ↓ 78 %

To compare the percents, ignore the percent signs and plot the numbers on a number line.

The given numbers can finally be ordered from least to greatest.

78 % ↓ 78 % < 80 % ↓ \frac{4}{5} < 91 % ↓ 0.91 < 122 % ↓ 1 \frac{2}{9}

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Catch-Up and Review

Comparing Math Homework Progress

Can Any Two Numbers Be Compared?

Comparing Two Numbers in the Same Form

Comparing Two Numbers in Decimal and Percent Forms

Ordering Decimals and Percents From Least to Greatest

Hint

Solution

Comparing Fractions

Common Denominator

The Most Convenient Common Denominator

Least Common Denominator

Extra

Comparing Two Fractions

It All Started With a Ball

Hint

Solution

Analyzing Facts About Flowers

Hint

Solution

Fraction

Decimal Number

Mixed Number

Comparison

Who Is Closer to Finishing Their Homework?

Hint

Solution

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