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}

Order the given percents and decimals from least to greatest.

41 %, 0.17, 29 %, 0.35

Let's start by analyzing the given list of numbers. 41 %, 0.17, 29 %, 0.35 There are two percents and two decimal numbers. Let's rewrite them so that all of the numbers are in the same form so that we can easily compare them. Percents:& 41 %, 29 % Decimals:& 0.17, 0.35 Let's rewrite the decimals as percents. For this, multiply the decimals by 100 and add a percent sign. We will start with 0.17. Move the decimal point two places to the right to multiply it by 100.

As we can see, 0.17 is equivalent to 17 %. 0.17* 100=17 % We can rewrite 0.35 into a percent in a similar manner. Let's move the decimal point two places to the right and add a percent sign.

We can see that 0.35 is equal to 35 %. 0.35* 100=35 % All the numbers are written in percent form now. cccc 41 % & 0.17 & 29 % & 0.35 [0.2cm] ↓ & ↓ & ↓ & ↓ [0.2cm] 41 % & 17 % & 29 % & 35 % Let's plot these numbers on a number line to help us order them from least to greatest. Remember that the farther to the right a number is located on the line, the greater it is.

Finally, the numbers can finally be ordered from the least to the greatest. ccccccc 17 % &< & 29 % &< & 35 % &< & 41 % [0.2cm] ↓ & & ↓ & & ↓ & & ↓ [0.2cm] 0.17, & & 29 %, & & 0.35, & & 41 %

Order the given fractions and percents from least to greatest.

\frac{2}{5}, 46 %, \frac{3}{8}, 38 %

We are given two fractions and two percents. 2/5, 46 %, 3/8, 38 % In order to compare these numbers, we need to rewrite them to be in the same form. Let's rewrite the fractions into percents. We will start with 25. First, divide the numerator by the denominator by using long division.

Next, multiply the decimal number by 100 and add a percent sign. 0.4* 100=40 % We can rewrite 38 as a percent in a similar fashion. Let's divide 3 by 8.

Now, multiply the decimal number by 100 and add a percent sign. 0.375* 100=37.5 % As we can see, 38 is equivalent to 37.5 %. All the numbers are now written in percent form! cccc 2/5, & 46 %, & 3/8, & 38 % [0.2cm] ↓ & ↓ & ↓ & ↓ [0.2cm] 40 %, & 46 %, & 37.5 %, & 38 % Finally, we can order the number from least to greatest by comparing their percent form. cccc 37.5 % &< & 38 % &< & 40 % &< & 46 % [0.2cm] ↑ & & ↑ & & ↑ & & ↑ [0.2cm] 3/8, & & 38 %, & & 2/5, & & 46 %

Tearrik needs to compare the speeds of different vehicles and animals. He wrote their speeds as they compare to $100$ kilometers per hour, which he set as $100 % .$

The speed of: 1) horse - 48 percent; 2) railway car - 227 percent; 3) tiger - 0.65; 4) racing car - 3.1

Order the vehicles and animals from fastest to slowest.

We are given four different numbers. 48 %, 227 %, 0.65, 3.1 Two of these numbers are percents and two are decimal numbers. We need to rewrite them so that they are in the same form in order to be able to compare them. Percents:& 48 %, 227 % Decimals:& 0.65, 3.1 Let's rewrite the numbers as decimals. To do this, divide the percents by 100 and remove the percent signs. We will start with 48 %. Move the decimal point two places to the left to divide it by 100.

As we can see, 48 % is equivalent to 0.48. 48 %÷ 100=0.48 We can rewrite 227 % as a percent in a similar manner. Let's move the decimal point two places to the left.

We can see that 227 % is the same as 2.27. 227 %÷ 100=2.27 All the numbers are now written in decimal form! cccc 48 % & 227 % & 0.65 & 3.1 [0.2cm] ↓ & ↓ & ↓ & ↓ [0.2cm] 0.48 & 2.27 & 0.65 & 3.1 Finally, let's compare these numbers and order them from greatest to least. ccccccc 3.1 &> & 2.27 &> & 0.65 &> & 0.48 [0.2cm] ↓ & & ↓ & & ↓ & & ↓ [0.2cm] 3.1, & & 227 %, & & 0.65, & & 48 % [0.2cm] ↓ & & ↓ & & ↓ & & ↓ [0.2cm] Racing car & & Railway car & & Tiger & & Horse Therefore, the racing car is the fastest, followed by the railway car, the tiger, and, finally, the horse.

Tearrik and Ali are solving their math homework. They need to compare a percent and a mixed number.

However, their solutions are different. Who is correct?

The boys need to compare two numbers, a mixed number and a percent. 4 59 and 462 % Let's analyze each solution step by step.

Tearrik's Solution

Tearrik decides to rewrite the mixed number as a percent to be able to compare the numbers. First, he converts the mixed number into an improper fraction by applying the corresponding formula. a bc=a* c+b/c Let's check if he applied the formula correctly.

The improper fraction that corresponds to 4 59 is 419. Tearrik calculated this step correctly. Next, he divides the numerator by the denominator using long division.

The quotient is about 4.55. Next, Tearrik needs to multiply this decimal number by 100 and add a percent sign. 4.55* 100=455 % This means that the mixed number is equivalent to about 455 %. Finally, Tearrik compares the percents and concludes that 462 % is greater than 455 %. 455 % < 462 % Tearrik's solution is absolutely correct!

Ali's Solution

Let's now consider Ali's solution. His first step is the same as Tearrik's. He rewrites the mixed number as an improper fraction. 4 59=41/9 ✓ Next, Ali rewrites the fraction as a percent. For this, he divides the numerator by the denominator. 41/9=4.55 He rounds the number to one decimal place and writes 4.6 %. However, he does not realize that he only found the corresponding decimal, and not the percent. To find the percent, he would also need to multiply the result by 100. This means that this step is incorrect. 41/9≈ 4.6 % * Therefore, Ali's solution is incorrect. Only Tearrik is correct.

Compare and Order Fractions, Decimals, and Percents

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

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Hint

Solution

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Hint

Solution

Fraction

Decimal Number

Mixed Number

Comparison

Who Is Closer to Finishing Their Homework?

Hint

Solution

Tearrik's Solution

Ali's Solution

Compare and Order Fractions, Decimals, and Percents

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