Grade 4 · 3.NF.A.3 · 3.NF.A.3B

Simplifying a fraction to simplest form

Twelve of the eighteen squares on a paint chip card are blue. Is there a shorter name for the same amount, using fewer and bigger pieces?

You need

Greatest common factor and least common multipleGrades 5 to 6 Making an equivalent fraction by multiplyingGrade 4

Unlocks

A fraction times a fractionGrade 5 Scaling a ratio up or downMath 6

Where you are 4 in Equivalent fractions

Fractions, ratio and percent3

Fractions15

Equivalent fractions4

Seeing equivalent fractions in pictures and on the number line Making an equivalent fraction by multiplying Common denominators and the least common denominator

Simplifying a fraction to simplest form

When you reach for it

Defne, Petro and Snezana are sorting paint chip cards. A paint chip card is a strip of stiff paper printed with small squares of color, every square the same size. The three of them describe the cards in fractions: twelve of the eighteen squares on this card are blue.

That is true, but it is a mouthful. Twelve eighteenths is the same amount as two thirds, and two thirds is much easier to picture. In Making an equivalent fraction by multiplying you cut every piece of a fraction into smaller pieces, which multiplies the top and the bottom by the same number. This lesson runs that move backward. Gluing small pieces together into bigger ones divides the top and the bottom by the same number, and the fraction gets a shorter name for the same amount.

You reach for this move whenever a fraction is written with bigger numbers than it needs: to read an amount at a glance, to see that two fractions are equal, and to keep later work with fractions small. The lesson has three parts, one for each day:

Part 1: group the small pieces of a model into bigger pieces.
Part 2: the rule, and simplest form, reached in steps or in one step with the greatest common factor.
Part 3: check that a simplified fraction is equal, and see why the number you divide by must be a factor of both the top and the bottom.

The procedure

Part 1: group small pieces into bigger ones

Snezana holds a paint chip card with 18 small squares, in 3 rows of 6. The left 4 columns are shades of blue, so 3 \times 4 = 12 squares are blue, and the blue part of the card is \tfrac{12}{18}.

Figure 1The same card counted in three sizes of piece. Grouping in pairs gives \tfrac{6}{9} and grouping in sixes gives \tfrac{2}{3}, so \tfrac{12}{18} = \tfrac{6}{9} = \tfrac{2}{3}.

She draws a line around the squares in pairs, two side by side in each row. Each row of 6 squares makes 3 pairs, so the card holds 3 \times 3 = 9 pairs. Each row has 4 blue squares, which make 2 blue pairs, so the card has 3 \times 2 = 6 blue pairs. Every pair is all blue or all green, so a pair is a fair piece to count. The blue part is 6 of the 9 pairs:

\frac{12}{18} = \frac{6}{9}

Then she groups bigger still, two whole columns at a time. Each group is 2 columns of 3 squares, which is 6 squares. The 18 squares make 18 \div 6 = 3 groups, and the 12 blue squares make 12 \div 6 = 2 groups. The blue part is 2 of the 3 groups:

\frac{12}{18} = \frac{6}{9} = \frac{2}{3}

No paint moved. The blue covers exactly the same part of the card as before. Only the size of the pieces she counts changed.

Look at what grouping did to the numbers. With groups of 2, the whole went from 18 squares to 18 \div 2 = 9 pairs, and the blue went from 12 squares to 12 \div 2 = 6 pairs. With groups of 6, the whole became 18 \div 6 = 3 pieces and the blue became 12 \div 6 = 2. The size of a group is the number you divide by, and it divides the top and the bottom alike, because the groups cover the whole card, blue squares and green squares together.

This is the move from Making an equivalent fraction by multiplying, undone. There, cutting every piece into n smaller pieces multiplied both numbers by n. Here, gluing every n pieces into one bigger piece divides both numbers by n. Fewer pieces, each one bigger, same amount.

Part 2: the rule, and simplest form

The rule. A number that is a factor of two numbers is a common factor of them. For example, 2 is a common factor of 12 and 18, because 12 = 2 \times 6 and 18 = 2 \times 9. For any fraction \tfrac{a}{b} and any common factor n of a and b,

\frac{a}{b} = \frac{a \div n}{b \div n}

Here is why. Because n is a factor of both numbers, a = n \times c and b = n \times d for some whole numbers c and d, and then c = a \div n and d = b \div n. The multiplying rule says that cutting every piece of \tfrac{c}{d} into n gives \tfrac{n \times c}{n \times d}, which is \tfrac{a}{b}. So \tfrac{a}{b} and \tfrac{c}{d} name the same amount: one is the other with its pieces cut, and the other is the first with its pieces glued. Dividing the top and the bottom by a common factor is called simplifying the fraction. Some books call it reducing.

Simplest form. Snezana could not group her card any further after \tfrac{2}{3}. The only number that is a factor of both 2 and 3 is 1, and groups of 1 change nothing. A fraction is in simplest form when its top and its bottom have no common factor greater than 1. Another name for this is lowest terms. So \tfrac{2}{3} is in simplest form, and \tfrac{6}{9} is not, because 3 is a factor of 6 and of 9.

To test a fraction, list the factors of one of its numbers, usually the smaller one, and ask whether any factor other than 1 also divides the other number. Factor pairs give the list. For \tfrac{6}{9}, the factors of 6 are 1, 2, 3 and 6. The factor 2 does not divide 9, but 3 does, so \tfrac{6}{9} can still be simplified.

In steps. You may divide by any common factor you spot, then look again. Snezana divided by 2 and then by 3:

\begin{aligned} \frac{12}{18} &= \frac{12 \div 2}{18 \div 2} = \frac{6}{9} \\ \frac{6}{9} &= \frac{6 \div 3}{9 \div 3} = \frac{2}{3} \end{aligned}

Petro starts with 3 instead: \tfrac{12 \div 3}{18 \div 3} = \tfrac{4}{6}, and then \tfrac{4 \div 2}{6 \div 2} = \tfrac{2}{3}. Different steps, same place. Every fraction has exactly one simplest form, so whichever common factors you use, you finish at the same fraction once no common factor greater than 1 is left.

In one step. The biggest common factor of two numbers is their greatest common factor, which Greatest common factor by listing finds by listing (the idea’s own star is Greatest common factor and least common multiple). For 12 and 18, the factors of 12 are 1, 2, 3, 4, 6 and 12. The ones that also divide 18 are 1, 2, 3 and 6, so the greatest common factor is 6. Dividing by it is Snezana’s grouping by two whole columns:

\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}

Dividing by the greatest common factor always finishes in one step. Suppose the new top and bottom, 2 and 3, still shared some factor k greater than 1. Then 12 = 6 \times 2 and 18 = 6 \times 3 would both have 6 \times k as a factor, and 6 \times k is bigger than 6. That cannot happen, because 6 was the greatest common factor. The same reasoning works for any fraction. Dividing by a smaller common factor is never wrong; it just may leave work to do, as dividing by 2 left \tfrac{6}{9}.

Part 3: check the answer, and why only a common factor works

Checking that the new fraction is equal. Simplifying undoes cutting, so cutting again must give back what you started with. If \tfrac{2}{3} came from dividing by 6, then multiplying its top and bottom by 6 must give \tfrac{12}{18}, and it does: 2 \times 6 = 12 and 3 \times 6 = 18. A number line shows the same thing.

Figure 2Every third holds 6 eighteenths, so twelve eighteenths and two thirds are the same point.

Each third of the space from 0 to 1 holds 18 \div 3 = 6 eighteenths. The point \tfrac{2}{3} is 2 thirds from 0, which is 2 \times 6 = 12 eighteenths. The two names land on one point.

Checking that it is simplest. Test the new top and bottom for a common factor greater than 1, as in Part 2. For \tfrac{2}{3}, the factors of 2 are 1 and 2, and 2 does not divide 3.

Why the number must be a factor of both. Defne has a color strip of 18 chips in a row, and the left 15 are orange. She tries grouping the chips in pairs.

Figure 3Pairs do not fit \tfrac{15}{18}, because one pair is half shaded. Groups of 3 fit, and \tfrac{15}{18} = \tfrac{5}{6}.

The strip makes 18 \div 2 = 9 pairs. But 15 is odd, so the orange chips do not fill a whole number of pairs: 7 pairs are all orange, and the eighth pair has one orange chip and one white one. That pair is neither an orange piece nor a white piece, so there is no whole number of orange pairs to count. The number 2 is a factor of 18 but not of 15, and that is exactly why it fails. Groups of 3 do work, because 3 is a factor of both: 18 \div 3 = 6 groups, and 15 \div 3 = 5 of them are all orange, so

\frac{15}{18} = \frac{15 \div 3}{18 \div 3} = \frac{5}{6}

The top and the bottom must also be divided by the same number. One size of group covers the whole strip, so one number divides both counts. Dividing the top of \tfrac{12}{18} by 2 and the bottom by 3 would give \tfrac{6}{6}, which is 1, the whole card, but only part of Snezana’s card is blue.

Worked examples

Examples 1 and 2 belong to Part 1. Examples 3 to 5 belong to Part 2. Examples 6 and 7 belong to Part 3.

Example 1

Pairs on a color strip

Petro has a color strip with 14 chips in a row. The 6 chips at the left end are purple. He groups the chips in pairs. Write the purple part of the strip as two equivalent fractions.

Solution

Before grouping, 6 of the 14 chips are purple, so the purple part is \tfrac{6}{14}.

The whole strip of 14 chips makes 14 \div 2 = 7 pairs. The 6 purple chips sit together at the left, and 6 is even, so they fill 6 \div 2 = 3 pairs exactly, with no pair half purple. The purple part is 3 of the 7 pairs:

\frac{6}{14} = \frac{6 \div 2}{14 \div 2} = \frac{3}{7}

The factors of 3 are 1 and 3, and 3 does not divide 7, so \tfrac{3}{7} is in simplest form. Check by cutting back: 3 \times 2 = 6 and 7 \times 2 = 14.

Example 2

Two ways to group one card

Defne’s paint chip card has 20 squares in 5 rows of 4. The top 2 rows are orange. Write the orange part as a fraction, then group the squares (a) in pairs along each row, and (b) one whole row at a time. Which grouping reaches simplest form?

Solution

The orange squares are 2 rows of 4, so 2 \times 4 = 8 of the 20 squares, which is \tfrac{8}{20}.

(a) Each row of 4 squares makes 2 pairs, so the card has 5 \times 2 = 10 pairs and the orange rows have 2 \times 2 = 4 pairs. That is \tfrac{4}{10}, from dividing both numbers by 2.

(b) Each row is one piece of 4 squares. The card is 5 rows and 2 are orange, so the orange part is \tfrac{2}{5}, from dividing both numbers by 4.

\frac{8}{20} = \frac{4}{10} = \frac{2}{5}

The pairs did not finish the job: 4 and 10 are both even, so \tfrac{4}{10} still has the common factor 2. The rows did. The factors of 2 are 1 and 2, and 2 does not divide 5, so \tfrac{2}{5} is in simplest form. Bigger groups made fewer, bigger pieces and went further in one step.

Example 3

Simplifying in steps

A rack holds 42 paint chip cards, and 30 of them show shades of red. Write the red part of the rack as a fraction and simplify it in steps.

Solution

The red part is \tfrac{30}{42}. Both numbers are even, so 2 is a common factor:

\frac{30}{42} = \frac{30 \div 2}{42 \div 2} = \frac{15}{21}

Look again. The factors of 15 are 1, 3, 5 and 15. The factor 3 divides 21, because 3 \times 7 = 21, so divide again:

\frac{15}{21} = \frac{15 \div 3}{21 \div 3} = \frac{5}{7}

Now the factors of 5 are 1 and 5, and 5 does not divide 7. No common factor greater than 1 is left, so \tfrac{5}{7} is the simplest form. The two steps divided by 2 and then by 3, which is the same as dividing once by 2 \times 3 = 6, and 6 is the greatest common factor of 30 and 42.

Example 4

One step with the greatest common factor

Simplify \tfrac{28}{42} in one step.

Solution

Find the greatest common factor. The factors of 28 are 1, 2, 4, 7, 14 and 28. Test each one on 42:

2 divides 42, since 2 \times 21 = 42.
4 does not, since 4 \times 10 = 40 and 4 \times 11 = 44.
7 divides 42, since 7 \times 6 = 42.
14 divides 42, since 14 \times 3 = 42.
28 does not, since 28 \times 1 = 28 and 28 \times 2 = 56.

The common factors are 1, 2, 7 and 14, and the greatest is 14. Divide by it:

\frac{28}{42} = \frac{28 \div 14}{42 \div 14} = \frac{2}{3}

Check by cutting back: 2 \times 14 = 28 and 3 \times 14 = 42. Working in steps would have taken two: dividing by 2 gives \tfrac{14}{21}, and dividing that by 7 gives \tfrac{2}{3}.

Example 5

A fraction greater than one

Petro is cutting color strips for a display. One display board takes 30 strips, and he has cut 45. Write the number of boards he has cut strips for as a fraction, in simplest form.

Solution

He has 45 strips and a board is 30, so he has \tfrac{45}{30} of a board’s worth.

The factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30. The ones that also divide 45 are 1, 3, 5 and 15, so the greatest common factor is 15:

\frac{45}{30} = \frac{45 \div 15}{30 \div 15} = \frac{3}{2}

The factors of 2 are 1 and 2, and 2 does not divide 3, so \tfrac{3}{2} is in simplest form. He has cut enough strips for \tfrac{3}{2} boards, one and a half. The rule does not care that the top is bigger than the bottom; simplest form is about common factors, not about size. In steps it also works: divide by 5 to get \tfrac{9}{6}, then by 3 to get \tfrac{3}{2}.

Example 6

Checking an answer

On a large sample board, 36 of the 60 chips are gray or beige. Petro writes \tfrac{36}{60} = \tfrac{6}{10}. Is his fraction equal to \tfrac{36}{60}? Is it in simplest form?

Solution

Equal: he divided both numbers by 6, and 6 is a factor of both, since 36 \div 6 = 6 and 60 \div 6 = 10. Check by cutting back: 6 \times 6 = 36 and 10 \times 6 = 60. So \tfrac{6}{10} is equal to \tfrac{36}{60}.

Simplest: no. Both 6 and 10 are even, so 2 is still a common factor, and \tfrac{6}{10} = \tfrac{3}{5}. Petro divided by a common factor, but not the greatest one. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36, and those that also divide 60 are 1, 2, 3, 4, 6 and 12. The greatest common factor is 12, and

\frac{36}{60} = \frac{36 \div 12}{60 \div 12} = \frac{3}{5}

Petro’s answer was correct but not finished. Dividing by 6 and then by 2 is dividing by 6 \times 2 = 12 in all.

Example 7

The same amount on two cards

Defne’s card has 32 squares, and 20 of them are green. Snezana’s card has 40 squares, and 25 of them are green. Is the same fraction of each card green?

Solution

Write each card’s green part in simplest form.

Defne: \tfrac{20}{32}. The factors of 20 are 1, 2, 4, 5, 10 and 20, and the ones that divide 32 are 1, 2 and 4. The greatest common factor is 4, so \tfrac{20}{32} = \tfrac{20 \div 4}{32 \div 4} = \tfrac{5}{8}.

Snezana: \tfrac{25}{40}. The factors of 25 are 1, 5 and 25, and the ones that divide 40 are 1 and 5. The greatest common factor is 5, so \tfrac{25}{40} = \tfrac{25 \div 5}{40 \div 5} = \tfrac{5}{8}.

Both cards are \tfrac{5}{8} green, so the same fraction of each card is green. Neither fraction comes from the other by dividing, since 25 is bigger than 20 and 20 is not a whole number times 25. They meet at their simplest form instead. Because every fraction has exactly one simplest form, two fractions are equal exactly when their simplest forms match.

Boundaries

A common factor only. The number you divide by must divide the top and the bottom with nothing left over. Otherwise some group would be partly counted and partly not, as with the pairs on the strip of 15 orange chips, and the new top or bottom would not be a whole number.
The same number on top and bottom. One size of group covers the whole, so one number divides both counts. Two different numbers name a different amount.
Not zero, and 1 does nothing. Groups of 0 pieces make no pieces at all, and division by 0 has no meaning. Groups of 1 leave every piece as it was, so dividing by 1 gives back the fraction you started with.
The same whole. \tfrac{12}{18} of a card and \tfrac{2}{3} of the same card are the same amount of paint. Simplifying never changes which whole the fraction is about.
Simplest form is about common factors, not size. \tfrac{3}{2} is in simplest form although it is greater than 1. Simplest form is defined for a fraction whose top and bottom are whole numbers and whose bottom is not 0.

Where this goes next

Simplest form is how fraction answers are usually written, so this move ends a great deal of later work. When you multiply two fractions, you can even simplify before you multiply and keep every number small (A fraction times a fraction).

Simplifying also helps you compare. Two fractions are equal exactly when their simplest forms match, and when they are not equal, renaming both with the same size of piece shows which is bigger (Common denominators and the least common denominator). The same idea reaches beyond fractions: a comparison such as 12 blue to 18 green can be scaled down the same way, to 2 to 3 (Scaling a ratio up or down).

Exercises

Part 1: group the pieces

A color strip has 10 chips in a row, and the left 6 are blue. Group the chips in pairs. Write the blue part in tenths and in fifths.
A paint chip card has 15 squares in 5 rows of 3, and the top 2 rows are red. Make each row one piece. Write the red part two ways.
A color strip has 16 chips, and the left 4 are pink. Group the chips in fours. What fraction of the strip is pink, in sixteenths and in fourths?
A card has 21 squares in 3 rows of 7, and the top row is purple. Make each row one piece. Write the purple part two ways.
A color strip has 12 chips, and the left 10 are yellow. Group the chips in pairs. How many pairs are there, and how many are yellow?
A paint chip card has 24 squares in 4 rows of 6, and the left 3 columns are blue. Make each column one piece. How many pieces are there, and how many are blue? Is the new fraction in simplest form?
A color strip has 20 chips, and the left 16 are green. Write the green part using groups of 2, and then using groups of 4.

Part 2: the rule and simplest form

Divide the top and the bottom by the given number.

\tfrac{14}{35}, by 7
\tfrac{27}{36}, by 9
Which of these are in simplest form: \tfrac{7}{12}, \tfrac{15}{25}, \tfrac{11}{33}, \tfrac{8}{9}? For each one that is not, name a common factor.

Write each fraction in simplest form.

\tfrac{40}{56}, in steps. Then say which single number would have done it in one step.
\tfrac{18}{30}
\tfrac{35}{50}
Find the greatest common factor of 42 and 56, and use it to simplify \tfrac{42}{56}.
\tfrac{48}{36}
\tfrac{60}{100}
Defne simplifies \tfrac{32}{40} by dividing by 2 three times. Petro does it by dividing once. What number did Petro divide by, and what answer do both get?

Part 3: checking, and the common factor

Petro says \tfrac{27}{45} = \tfrac{3}{5}. Check by multiplying back. Is \tfrac{3}{5} in simplest form?
Snezana says \tfrac{24}{36} = \tfrac{4}{6}. Is it equal? Is it in simplest form?
Defne writes \tfrac{14}{16} = \tfrac{7}{16}, saying “I divided the top by 2.” Is she right? What is the correct simplest form?
Can \tfrac{9}{14} be simplified? Explain using factors.
Are \tfrac{15}{35} and \tfrac{12}{28} equal? Simplify each to decide.
A color strip has 21 chips, and the left 14 are orange. Can the chips be grouped in pairs to simplify? In groups of 7? Write the orange part in simplest form.

Reasoning

Explain why dividing the top and the bottom by their greatest common factor gives simplest form in one step, while dividing by a smaller common factor might not.
Petro says: “To simplify \tfrac{12}{18}, I divide the top by 6 and the bottom by 9, and get \tfrac{2}{2}.” Explain what is wrong, using Snezana’s card.

Answers

\tfrac{6}{10} = \tfrac{3}{5}. The strip makes 10 \div 2 = 5 pairs, and the blue chips make 6 \div 2 = 3 pairs.
\tfrac{6}{15} = \tfrac{2}{5}. The red rows hold 2 \times 3 = 6 of the 15 squares. As rows, the card is 5 pieces and 2 are red, which divides both numbers by 3.
\tfrac{4}{16} = \tfrac{1}{4}. The strip makes 16 \div 4 = 4 groups, and the pink chips fill 4 \div 4 = 1 of them.
\tfrac{7}{21} = \tfrac{1}{3}. The purple row is 7 of the 21 squares. As rows, the card is 3 pieces and 1 is purple, which divides both numbers by 7.
6 pairs, 5 yellow, so \tfrac{10}{12} = \tfrac{5}{6}. The strip makes 12 \div 2 = 6 pairs, and 10 \div 2 = 5 of them are yellow.
6 pieces, 3 blue, so \tfrac{12}{24} = \tfrac{3}{6}. The blue columns hold 4 \times 3 = 12 of the 24 squares, and each column is 4 squares, so this divides both numbers by 4. Not simplest: 3 divides both 3 and 6, and \tfrac{3}{6} = \tfrac{1}{2}.
\tfrac{8}{10} with pairs, and \tfrac{4}{5} with groups of 4. Pairs: 20 \div 2 = 10 and 16 \div 2 = 8. Fours: 20 \div 4 = 5 and 16 \div 4 = 4. So \tfrac{16}{20} = \tfrac{8}{10} = \tfrac{4}{5}.
\tfrac{2}{5}, since 14 \div 7 = 2 and 35 \div 7 = 5.
\tfrac{3}{4}, since 27 \div 9 = 3 and 36 \div 9 = 4.
\tfrac{7}{12} and \tfrac{8}{9} are in simplest form. \tfrac{15}{25} is not (5 divides both), and \tfrac{11}{33} is not (11 divides both, since 11 \times 3 = 33). For \tfrac{7}{12}: 7 does not divide 12. For \tfrac{8}{9}: the factors 2, 4, 8 of 8 do not divide 9.
\tfrac{5}{7}. Dividing by 2 three times: \tfrac{40}{56} = \tfrac{20}{28} = \tfrac{10}{14} = \tfrac{5}{7}. The single number is 2 \times 2 \times 2 = 8, the greatest common factor.
\tfrac{3}{5}. The greatest common factor of 18 and 30 is 6: 18 \div 6 = 3 and 30 \div 6 = 5.
\tfrac{7}{10}. The greatest common factor of 35 and 50 is 5: 35 \div 5 = 7 and 50 \div 5 = 10.
14, and \tfrac{42}{56} = \tfrac{3}{4}. The factors of 42 are 1, 2, 3, 6, 7, 14, 21, 42; of these, 1, 2, 7 and 14 divide 56. Then 42 \div 14 = 3 and 56 \div 14 = 4.
\tfrac{4}{3}. The greatest common factor of 48 and 36 is 12: 48 \div 12 = 4 and 36 \div 12 = 3.
\tfrac{3}{5}. The greatest common factor of 60 and 100 is 20: 60 \div 20 = 3 and 100 \div 20 = 5.
Petro divided by 8, since 2 \times 2 \times 2 = 8, and both get \tfrac{4}{5}: 32 \div 8 = 4 and 40 \div 8 = 5.
Correct, and simplest. 27 \div 9 = 3 and 45 \div 9 = 5; multiplying back, 3 \times 9 = 27 and 5 \times 9 = 45. The factors of 3 are 1 and 3, and 3 does not divide 5.
Equal, but not simplest. She divided both by 6: 4 \times 6 = 24 and 6 \times 6 = 36. But 4 and 6 share the factor 2, so the simplest form is \tfrac{2}{3}, from the greatest common factor 12.
No. \tfrac{7}{16} is only half as much as \tfrac{14}{16}, because she halved the count of pieces without making the pieces bigger. Dividing both by 2 gives \tfrac{7}{8}, which is the simplest form.
No, it is already in simplest form. The factors of 9 are 1, 3 and 9, and neither 3 nor 9 divides 14.
Yes. \tfrac{15}{35} = \tfrac{3}{7} (dividing by 5) and \tfrac{12}{28} = \tfrac{3}{7} (dividing by 4). Both simplest forms are \tfrac{3}{7}.
Not in pairs: 2 is a factor of 14 but not of 21, so the strip does not split into whole pairs. Groups of 7 work: 21 \div 7 = 3 groups, and 14 \div 7 = 2 are orange. The simplest form is \tfrac{2}{3}.
Call the greatest common factor g, and say \tfrac{a}{b} divides to \tfrac{c}{d}, so a = g \times c and b = g \times d. If c and d still shared a factor k greater than 1, then g \times k would be a factor of both a and b, and it is bigger than g. That is impossible, since g is the greatest, so c and d have no common factor greater than 1. A smaller common factor leaves part of g behind: dividing \tfrac{12}{18} by 2 gives \tfrac{6}{9}, and the leftover 3 is still common to 6 and 9.
\tfrac{2}{2} is 1, the whole card, but only 12 of the 18 squares are blue, so the amount changed. Grouping the blue squares in sixes and the whole card in nines uses two sizes of piece at once, and a fraction must count one size of piece. Use one number for both: \tfrac{12 \div 6}{18 \div 6} = \tfrac{2}{3}.