Free math worksheets

- The two bars represent wholes of the same size. - Compare how the halfway point of bar A lines up with the fourth-sized parts in bar B. - The equivalent fraction must name the same amount using fourths.

1. Bar A has \(1\) of \(2\) equal parts shaded, so it represents \(\frac{1}{2}\). 2. In bar B, the same amount covers \(2\) of the \(4\) equal parts. 3. Therefore, \(\frac{1}{2}=\frac{2}{4}\).

Shade \(2\) parts; \(\frac{2}{4}\).

- Compare the two denominators. - Ask how the number of equal parts changed from the first fraction to the second. - Equivalent fractions must scale the numerator and denominator in the same way.

1. The denominator changes from \(3\) to \(6\), which doubles the number of equal parts. 2. To keep the same fraction of the whole, double the numerator as well. 3. The missing numerator is \(4\), so \(\frac{2}{3}=\frac{4}{6}\).

\(4\)

- Check whether the numerator and denominator share a factor greater than \(1\). - Test a few small factors for each fraction. - A fraction is in simplest form only when no whole number greater than \(1\) divides both parts.

1. In a), \(6\) and \(8\) are both divisible by \(2\), so \(\frac{6}{8}\) can be simplified. 2. In b), \(5\) and \(12\) have no common factor greater than \(1\), so \(\frac{5}{12}\) is in simplest form. 3. In c), \(4\) and \(10\) are both divisible by \(2\), so \(\frac{4}{10}\) can be simplified. 4. In d), \(9\) and \(12\) are both divisible by \(3\), so \(\frac{9}{12}\) can be simplified. 5. Therefore, only \(\frac{5}{12}\) is in simplest form.

b) \(\frac{5}{12}\)

- Look for the multiplication or division that changes one known part of a fraction into the matching part of the other fraction. - Whatever you do to the numerator, do the same to the denominator. - If the scale is not obvious, simplify one fraction first and then rebuild an equivalent fraction with the needed denominator.

1. For a), \(4\) is multiplied by \(3\) to get \(12\), so multiply \(3\) by \(3\): \(3\times3=9\). 2. For b), \(2\) is multiplied by \(2\) to get \(4\), so multiply \(5\) by \(2\): \(5\times2=10\). 3. For c), \(\frac{6}{12}=\frac{1}{2}\). To write one half with denominator \(8\), multiply both parts of \(\frac{1}{2}\) by \(4\): \(\frac{4}{8}\).

a) \(9\) b) \(10\) c) \(4\)

- Compare the old numerator with the new numerator. - Check whether the denominator changed by the same factor. - Equivalent fractions are made by multiplying both parts by the same nonzero whole number.

1. In a), both parts of \(\frac{1}{4}\) are multiplied by \(2\), so the scale factor is \(2\). 2. In b), both parts of \(\frac{2}{4}\) are multiplied by \(3\), so the scale factor is \(3\). 3. In c), both parts of \(\frac{1}{2}\) are multiplied by \(4\), so the scale factor is \(4\).

a) \(2\) b) \(3\) c) \(4\)

- Look at how the denominator changes in each pair. - Apply the same multiplication to the numerator. - Check that each completed fraction has the same value as the original fraction.

1. For a), the denominator is multiplied by \(2\), so \(3\times2=6\). 2. For b), the denominator is multiplied by \(2\), so \(5\times2=10\). 3. For c), the denominator is multiplied by \(2\), so \(3\times2=6\). 4. For d), the denominator is multiplied by \(5\), so \(5\times5=25\).

a) \(6\) b) \(10\) c) \(6\) d) \(25\)

- Look at how the smaller denominator can be changed into the larger denominator. - Apply the same multiplication to the numerator. - Once the denominators match, compare the numerators.

1. For a), rewrite \(\frac{5}{6}\) with denominator \(12\): \(\frac{5\times2}{6\times2}=\frac{10}{12}\). The fractions are equivalent. 2. For b), rewrite \(\frac{3}{4}\) with denominator \(8\): \(\frac{3\times2}{4\times2}=\frac{6}{8}\). Since \(\frac{6}{8}\ne\frac{5}{8}\), the fractions are not equivalent.

a) Equivalent, because \(\frac{5}{6}=\frac{10}{12}\). b) Not equivalent, because \(\frac{6}{8}\ne\frac{5}{8}\).

- Reducing by a given number means dividing both the numerator and denominator by that number. - Both divisions must give whole numbers. - Test the numerator and denominator separately before writing the new fraction.

1. For a), \(8\div4=2\) and \(12\div4=3\), so the result is \(\frac{2}{3}\). 2. For b), \(9\div3=3\) and \(12\div3=4\), so the result is \(\frac{3}{4}\). 3. For c), \(6\div2=3\) and \(10\div2=5\), so the result is \(\frac{3}{5}\). 4. For d), \(5\div5=1\) and \(10\div5=2\), so the result is \(\frac{1}{2}\). 5. For e), neither \(8\) nor \(12\) is divisible by \(5\), so the reduction is not possible.

a) \(\frac{2}{3}\) b) \(\frac{3}{4}\) c) \(\frac{3}{5}\) d) \(\frac{1}{2}\) e) not possible

- Use the result of the first division as the fraction for the second division. - Think about what one divisor would have the same effect as dividing by \(2\) and then by \(3\). - Check your one-step divisor on both the numerator and denominator.

1. Divide by \(2\): \(\frac{18}{12}\rightarrow\frac{9}{6}\). 2. Divide by \(3\): \(\frac{9}{6}\rightarrow\frac{3}{2}\). Since \(3\) and \(2\) have no common factor greater than \(1\), \(\frac{3}{2}\) is in simplest form. 3. The two divisors combine to \(2\times3=6\). 4. Check: \(18\div6=3\) and \(12\div6=2\).

a) \(\frac{3}{2}\) b) \(6\)

- Compare the known numerators or denominators in each equation. - Apply the same multiplication or division to both parts of a fraction. - If a direct scale factor is not obvious, simplify one fraction first.

1. For a), \(2\) is multiplied by \(2\) to get \(4\), so \(5\times2=10\). 2. For b), \(\frac{4}{8}=\frac{1}{2}\). With denominator \(6\), one half is \(\frac{3}{6}\). 3. For c), divide both parts of \(\frac{6}{8}\) by \(2\) to get \(\frac{3}{4}\). 4. For d), \(4\) is multiplied by \(3\) to get \(12\), so \(3\times3=9\).

a) \(10\) b) \(3\) c) \(4\) d) \(9\)

- Put the number in the part over the total number in the whole group. - Look for a whole number that divides both parts of the fraction. - Keep simplifying until the numerator and denominator have no common factor greater than \(1\).

1. For a), \(\frac{4}{10}=\frac{2}{5}\). 2. For b), \(\frac{3}{12}=\frac{1}{4}\). 3. For c), \(\frac{6}{8}=\frac{3}{4}\). 4. For d), \(\frac{2}{6}=\frac{1}{3}\).

a) \(\frac{2}{5}\) b) \(\frac{1}{4}\) c) \(\frac{3}{4}\) d) \(\frac{1}{3}\)

- Look for a common factor of the numerator and denominator. - Divide both parts of the fraction by the same factor. - Stop when no whole number greater than \(1\) divides both parts.

1. For a), divide the numerator and denominator by \(2\): \(\frac{6}{8}=\frac{3}{4}\). 2. For b), divide by \(4\): \(\frac{4}{12}=\frac{1}{3}\). 3. For c), divide by \(2\): \(\frac{6}{10}=\frac{3}{5}\).

a) \(\frac{3}{4}\) b) \(\frac{1}{3}\) c) \(\frac{3}{5}\)

- Simplify each fraction separately. - Use the same divisor on the numerator and denominator of one fraction. - Compare the two simplest forms.

1. Divide \(9\) and \(12\) by \(3\): \(\frac{9}{12}=\frac{3}{4}\). 2. Divide \(6\) and \(8\) by \(2\): \(\frac{6}{8}=\frac{3}{4}\). 3. Both fractions simplify to \(\frac{3}{4}\), so they are equivalent.

Yes. Both fractions simplify to \(\frac{3}{4}\).

- Identify the total number of equal parts and the number selected. - Find a number that divides both the numerator and denominator. - Imagine removing some dividing lines to group small slices into larger equal pieces.

1. Three of the \(12\) slices are on the plate, so the fraction is \(\frac{3}{12}\). 2. Divide the numerator and denominator by \(3\): \(\frac{3}{12}=\frac{1}{4}\). 3. Grouping every \(3\) small slices makes \(4\) larger equal pieces. The \(3\) slices on the plate make exactly \(1\) of those \(4\) pieces.

a) \(\frac{3}{12}\) b) \(\frac{1}{4}\) c) The \(3\) small slices together make one-fourth of the whole pizza.

- A fraction is in simplest form when its numerator and denominator share no factor greater than \(1\). - Compare the factors of each numerator with the factors of its denominator. - If you find a common factor, divide both parts by that same number.

1. The factors of \(5\) are \(1\) and \(5\). The factors of \(12\) are \(1,2,3,4,6,12\). Their only common factor is \(1\), so \(\frac{5}{12}\) is already in simplest form. 2. The numerator \(6\) and denominator \(8\) share the factor \(2\). Divide both by \(2\): \(\frac{6}{8}=\frac{3}{4}\).

\(\frac{5}{12}\) cannot be simplified. \(\frac{6}{8}\) simplifies to \(\frac{3}{4}\).

- Find the least common multiple of \(10\) and \(25\). - Determine the scale factor from each original denominator to the common denominator. - Multiply each numerator by the same factor used for its denominator.

1. The least common multiple of \(10\) and \(25\) is \(50\), so the least common denominator is \(50\). 2. Rewrite \(\frac{3}{10}\): \(\frac{3\times5}{10\times5}=\frac{15}{50}\). 3. Rewrite \(\frac{4}{25}\): \(\frac{4\times2}{25\times2}=\frac{8}{50}\).

The least common denominator is \(50\). The fractions are \(\frac{15}{50}\) and \(\frac{8}{50}\).

- The stated divisor must divide both the numerator and denominator evenly. - Check each part of each fraction separately. - After dividing, see whether the resulting fraction is already in simplest form.

1. In a), \(6\) is divisible by \(6\), but \(8\) is not. 2. In b), both \(6\) and \(12\) are divisible by \(6\): \(\frac{6}{12}=\frac{1}{2}\). 3. In c), \(12\) is divisible by \(6\), but \(8\) is not. 4. Therefore, only b) can be simplified by \(6\), and \(\frac{1}{2}\) is in simplest form.

b) \(\frac{6}{12}=\frac{1}{2}\)

- Simplify each fraction when possible. - Fractions that reduce to the same value are equivalent. - You can also check whether the numerator and denominator were multiplied by the same factor.

1. Card A shows \(\frac{1}{2}\). 2. Simplify card C: \(\frac{3}{6}=\frac{1}{2}\). 3. Simplify card D: \(\frac{5}{10}=\frac{1}{2}\). 4. Card B shows \(\frac{3}{4}\), which is not equal to \(\frac{1}{2}\). 5. Therefore, cards A, C, and D are equivalent.

Cards A, C, and D are equivalent; each represents \(\frac{1}{2}\).

- For part a), compare the numerators because the denominators are the same. - A fraction equal to one-half has a numerator that is half its denominator. - For part c), imagine dividing each fourth into two equal pieces.

1. The fractions in part a) have the same denominator. Since \(3 < 7\), \(\frac{3}{10} < \frac{7}{10}\). 2. A fraction is equivalent to \(\frac{1}{2}\) when its numerator is half its denominator. This is true for \(\frac{2}{4}\) and \(\frac{4}{8}\). 3. Multiplying both the numerator and denominator of \(\frac{2}{4}\) by \(2\) gives \(\frac{4}{8}\). Dividing the same whole into twice as many equal parts requires twice as many parts to represent the same amount.

a) \(\frac{3}{10} < \frac{7}{10}\) b) \(\frac{2}{4}\) and \(\frac{4}{8}\) c) Both fractions represent one-half of the whole; multiplying the numerator and denominator of \(\frac{2}{4}\) by \(2\) gives \(\frac{4}{8}\).

- Multiply or divide the numerator and denominator by the same number. - Look at how the known numerator or denominator changed. - Remember that a fraction with the same numerator and denominator equals \(1\).

1. For part a), multiply the numerator and denominator by the same number: \(\frac{1}{2} = \frac{2}{4} = \frac{4}{8} = \frac{5}{10}\). 2. For part b), multiply the numerator and denominator of \(\frac{1}{4}\) by \(2\): \(\frac{1}{4} = \frac{2}{8}\). 3. For part c), the numerator is multiplied by \(2\), so multiply the denominator by \(2\): \(\frac{3}{4} = \frac{6}{8}\). 4. For part d), \(\frac{10}{10} = 1\), so the second fraction must also have equal numerator and denominator: \(\frac{2}{2}\).

a) \(\frac{1}{2} = \frac{2}{4} = \frac{4}{8} = \frac{5}{10}\) b) \(\frac{1}{4} = \frac{2}{8}\) c) \(\frac{3}{4} = \frac{6}{8}\) d) \(\frac{10}{10} = \frac{2}{2}\)

- Think about how many eighth-size pieces fit in one fourth-size piece. - Rewrite fourths as eighths by multiplying the numerator and denominator by the same number. - A quick sketch of the two pizzas may help.

1. Each piece of the first pizza is \(\frac{1}{4}\), so \(2\) pieces represent \(\frac{2}{4} = \frac{1}{2}\). 2. One fourth is equivalent to two eighths: \(\frac{1}{4} = \frac{2}{8}\). Therefore, \(2\) pieces of the second pizza make \(\frac{1}{4}\). 3. Rewrite \(\frac{3}{4}\) in eighths by multiplying the numerator and denominator by \(2\): \(\frac{3}{4} = \frac{6}{8}\). Therefore, \(6\) pieces should be packed.

a) \(\frac{2}{4}\), or \(\frac{1}{2}\) b) \(2\) pieces c) \(6\) pieces

- Equivalent fractions can be created by multiplying or dividing the numerator and denominator by the same number. - For part d), rewrite both fractions with denominator \(8\). - Once the denominators match, compare the numerators.

1. Multiply the numerator and denominator of \(\frac{1}{2}\) by \(4\): \(\frac{1}{2} = \frac{4}{8}\). 2. Multiply the numerator and denominator of \(\frac{3}{4}\) by \(2\): \(\frac{3}{4} = \frac{6}{8}\). 3. Divide the numerator and denominator of \(\frac{6}{8}\) by \(2\): \(\frac{6}{8} = \frac{3}{4}\). 4. Rewrite \(\frac{1}{2}\) as \(\frac{4}{8}\). Since \(5 > 4\), \(\frac{5}{8} > \frac{4}{8}\), so \(\frac{5}{8}\) is greater.

a) \(\frac{1}{2} = \frac{4}{8}\) b) \(\frac{3}{4} = \frac{6}{8}\) c) \(\frac{6}{8} = \frac{3}{4}\) d) \(\frac{5}{8}\) is greater because \(\frac{1}{2} = \frac{4}{8}\) and \(\frac{5}{8} > \frac{4}{8}\).

- How many cents make one whole dollar? - What does the denominator tell you about the number of equal parts in the whole? - If a dollar is divided into four equal amounts, how many cents are in each amount?

1. Because \(\$1 = 100\) cents, \(1\) cent is \(\frac{1}{100}\) of a dollar. 2. Similarly, \(18\) cents is \(\frac{18}{100}\) of a dollar, and \(73\) cents is \(\frac{73}{100}\) of a dollar. 3. Since \(100 \div 4 = 25\), \(25\) cents is one of four equal parts of a dollar. Equivalently, \(\frac{25}{100} = \frac{1}{4}\).

a) \(1\) cent is \(\frac{1}{100}\) of a dollar; \(18\) cents is \(\frac{18}{100}\); and \(73\) cents is \(\frac{73}{100}\). b) A dollar has \(100\) cents, and \(100 \div 4 = 25\). Therefore, \(25\) cents is \(\frac{1}{4}\) of a dollar.

- Think of one dollar as \(100\) equal cents. - The hundredths place in a dollar amount records cents. - For part b), make two different coin totals that both equal \(25\) cents.

1. One cent is one hundredth of a dollar, so it is \(\$0.01\). 2. Five cents is five hundredths of a dollar, so it is \(\$0.05\). 3. Forty cents is forty hundredths of a dollar, so it is \(\$0.40\). 4. \(\$0.25\) is \(25\) cents. One possible coin combination is one quarter. 5. Another possible combination is two dimes and one nickel.

a) \(\$0.01\), \(\$0.05\), \(\$0.40\) b) For example, one quarter; or two dimes and one nickel.

- Count all equal parts to find the denominator. - Count the shaded parts to find the numerator. - Simplify when the numerator and denominator share a common factor.

1. In a), \(6\) of \(8\) equal sectors are shaded: \(\frac{6}{8}=\frac{3}{4}\). 2. In b), \(8\) of \(12\) equal squares are shaded: \(\frac{8}{12}=\frac{2}{3}\). 3. In c), \(2\) of \(5\) equal parts are shaded, so the fraction is \(\frac{2}{5}\).

a) \(\frac{3}{4}\) b) \(\frac{2}{3}\) c) \(\frac{2}{5}\)

- Decide whether the question asks about the shaded pieces or the unshaded pieces. - Count the total number of equal pieces in the image. - Write the unshaded count over the total, then simplify.

1. The bar has \(12\) equal pieces. 2. Three pieces are shaded, so \(12-3=9\) pieces remain. 3. The remaining fraction is \(\frac{9}{12}=\frac{3}{4}\).

\(\frac{3}{4}\)

- For each figure, count the total number of equal parts to find the denominator. - Count the shaded parts to find the numerator. - Simplify each fraction, then compare the results.

1. In figure a), \(4\) of \(8\) equal parts are shaded, so the fraction is \(\frac{4}{8} = \frac{1}{2}\). 2. In figure b), \(6\) of \(12\) equal parts are shaded, so the fraction is \(\frac{6}{12} = \frac{1}{2}\). 3. In figure c), \(2\) of \(5\) equal parts are shaded, so the fraction is \(\frac{2}{5}\). 4. Therefore, figures a) and b) represent the same fraction, \(\frac{1}{2}\).

Figures a) and b) represent the same fraction. In simplest form, the fraction is \(\frac{1}{2}\).

- Count all spaces in the grid. - Count the blue-shaded spaces and the unshaded spaces separately. - Write each count over the total, then simplify.

1. The grid has \(2\) rows and \(5\) columns, so there are \(10\) spaces. 2. Four spaces are blue-shaded, so the filled fraction is \(\frac{4}{10}=\frac{2}{5}\). 3. Six spaces are unshaded, so the empty fraction is \(\frac{6}{10}=\frac{3}{5}\).

a) \(\frac{2}{5}\) b) \(\frac{3}{5}\)

- Read Ava's fraction directly from figure A. - Compare how many equal pieces the two same-size cakes use. - An equivalent fraction changes the numerator and denominator by the same factor.

1. Figure A shows \(3\) of \(4\) equal pieces shaded, so Ava ate \(\frac{3}{4}\) of her cake. 2. Figure B has \(8\) equal pieces. To write \(\frac{3}{4}\) in eighths, multiply the numerator and denominator by \(2\): \(\frac{3\times2}{4\times2}=\frac{6}{8}\). 3. Luis must eat \(6\) pieces, which is \(\frac{6}{8}\) of his cake.

a) \(\frac{3}{4}\) b) \(6\) pieces c) \(\frac{6}{8}\)

- Use the rows and columns in the image to find the total number of pieces. - Put each part count over the same total. - Simplify each fraction with a common factor.

1. The array has \(2\) rows and \(5\) columns, so the bar had \(2\times5=10\) pieces. 2. There are \(6\) orange-shaded pieces. The fraction remaining is \(\frac{6}{10}=\frac{3}{5}\). 3. There are \(4\) unshaded pieces. The fraction eaten is \(\frac{4}{10}=\frac{2}{5}\).

a) \(10\) pieces b) \(\frac{3}{5}\) c) \(\frac{2}{5}\)

- Count all the marbles shown. - Count the blue marbles. - Write the blue count over the total count and simplify.

1. There are \(10\) marbles in all. 2. Four marbles are blue, so the fraction is \(\frac{4}{10}\). 3. Divide the numerator and denominator by \(2\): \(\frac{4}{10}=\frac{2}{5}\).

\(\frac{2}{5}\)

- Count all equal squares in the grid. - Count the blue-shaded and unshaded squares separately. - Check whether each numerator shares a common factor greater than \(1\) with the denominator.

1. The grid has \(3\) rows and \(4\) columns, so it contains \(12\) squares. 2. Five squares are blue-shaded, so the shaded fraction is \(\frac{5}{12}\), which is already in simplest form. 3. Seven squares are unshaded, so the unshaded fraction is \(\frac{7}{12}\), which is already in simplest form.

a) \(\frac{5}{12}\) b) \(\frac{7}{12}\)

- Count the equal intervals from \(0\) to \(1\). - What fraction does one small interval represent? - Count intervals from \(0\) to determine each numerator. - Simplify each fraction completely.

1. The interval from \(0\) to \(1\) is divided into \(6\) equal parts, so each interval represents \(\frac{1}{6}\). 2. Point \(A\) is at the first tick, so \(A = \frac{1}{6}\). 3. Point \(B\) is at the second tick, so \(B = \frac{2}{6} = \frac{1}{3}\). 4. Point \(C\) is at the fourth tick, so \(C = \frac{4}{6} = \frac{2}{3}\).

\(A = \frac{1}{6}\), \(B = \frac{1}{3}\), and \(C = \frac{2}{3}\).

- Count the equal intervals from \(0\) to \(1\) to determine the denominator. - Each point represents a certain number of those intervals. - Check whether the numerator and denominator have a common factor.

1. The interval from \(0\) to \(1\) is divided into \(8\) equal parts, so each interval represents \(\frac{1}{8}\). 2. Point \(P\) is at the second tick: \(\frac{2}{8} = \frac{1}{4}\). 3. Point \(Q\) is at the fifth tick: \(\frac{5}{8}\). 4. Point \(R\) is at the sixth tick: \(\frac{6}{8} = \frac{3}{4}\).

\(P = \frac{1}{4}\), \(Q = \frac{5}{8}\), and \(R = \frac{3}{4}\).

- Count the total number of equal parts in the circle. - Count the shaded parts and write that fraction. - Simplify the fraction and compare it with one-half.

1. The circle is divided into \(4\) equal parts, and \(2\) parts are shaded. The shaded fraction is \(\frac{2}{4}\). 2. Simplify \(\frac{2}{4}\) by dividing the numerator and denominator by \(2\): \(\frac{2 \div 2}{4 \div 2} = \frac{1}{2}\). 3. Since \(\frac{2}{4} = \frac{1}{2}\), both students describe the same shaded amount and are correct.

Both Jordan and Mia are correct. The figure shows \(\frac{2}{4}\), and \(\frac{2}{4} = \frac{1}{2}\).

- Read the number of equal parts and shaded parts from the figure. - An equivalent fraction names the same amount using different numbers. - Look for a common factor of the numerator and denominator.

1. The image shows \(3\) of \(6\) equal triangles shaded, so the shaded fraction is \(\frac{3}{6}\). 2. Divide the numerator and denominator by \(3\): \(\frac{3}{6}=\frac{1}{2}\).

a) \(\frac{3}{6}\) b) \(\frac{1}{2}\)

- Count all equal sectors to find the denominator. - Count the green-shaded sectors for part a). - Count the unshaded sectors for part b), then simplify each fraction.

1. The circle is divided into \(12\) equal sectors, and \(9\) are green-shaded. 2. The fraction who said yes is \(\frac{9}{12}=\frac{3}{4}\). 3. The number of unshaded sectors is \(12-9=3\). The fraction who said no is \(\frac{3}{12}=\frac{1}{4}\).

a) \(\frac{3}{4}\) b) \(\frac{1}{4}\)

- Every group has a total that can be divided into fourths, so the total alone cannot decide the answer. - Compare the shaded count with one-fourth of the total in each group. - Check each candidate by writing and simplifying its shaded fraction.

1. In group a), \(2\) of \(8\) objects are shaded: \(\frac{2}{8}=\frac{1}{4}\). 2. In group b), \(4\) of \(12\) objects are shaded: \(\frac{4}{12}=\frac{1}{3}\). 3. In group c), \(4\) of \(16\) objects are shaded: \(\frac{4}{16}=\frac{1}{4}\). 4. In group d), \(6\) of \(20\) objects are shaded: \(\frac{6}{20}=\frac{3}{10}\). 5. Therefore, groups a) and c) show one-fourth shaded.

Groups a) and c).

- Count all the chocolates shown to find the denominator. - Count the gray chocolates to find the numerator. - Simplify by dividing the numerator and denominator by the same common factor.

1. The image shows \(10\) chocolates in all. 2. Four chocolates are shaded gray, so the dark-chocolate fraction is \(\frac{4}{10}\). 3. Divide the numerator and denominator by \(2\): \(\frac{4}{10}=\frac{2}{5}\).

\(\frac{2}{5}\)

- Read the number of equal pieces in Noah's pizza from the figure. - Find a fraction equivalent to \(\frac{1}{3}\) with that denominator. - The numerator of the equivalent fraction tells how many pieces Noah should eat.

1. Noah's pizza is divided into \(12\) equal pieces. 2. Write \(\frac{1}{3}\) as an equivalent fraction with denominator \(12\): \(\frac{1\times4}{3\times4}=\frac{4}{12}\). 3. Noah must eat \(4\) pieces.

Noah must eat \(4\) pieces.

- Count the total number of equal squares. - Count the green-shaded squares. - Write the shaded count over the total and simplify.

1. The grid has \(2\) rows and \(5\) columns, so it contains \(10\) equal squares. 2. Four squares are green-shaded, so the fraction is \(\frac{4}{10}\). 3. Divide the numerator and denominator by \(2\): \(\frac{4}{10}=\frac{2}{5}\).

Strawberries cover \(\frac{2}{5}\) of the garden bed.

- Count all paint pots shown. - Count the purple-shaded pots. - Write the shaded count over the total and simplify.

1. The image shows \(12\) paint pots in all. 2. Four pots are purple-shaded, so the fraction used is \(\frac{4}{12}\). 3. Divide the numerator and denominator by \(4\): \(\frac{4}{12}=\frac{1}{3}\).

\(\frac{1}{3}\)

- Count the total number of equal parts and the number of shaded parts. - Write the shaded count as the numerator and the total count as the denominator. - Divide the numerator and denominator by a common factor.

1. In figure a), the grid has \(3\times4=12\) equal squares. Nine are shaded, so the fraction is \(\frac{9}{12}\). Divide the numerator and denominator by \(3\): \(\frac{9}{12}=\frac{3}{4}\). 2. In figure b), the hexagon has \(6\) equal parts. Four are shaded, so the fraction is \(\frac{4}{6}\). Divide the numerator and denominator by \(2\): \(\frac{4}{6}=\frac{2}{3}\).

a) \(\frac{9}{12}=\frac{3}{4}\) b) \(\frac{4}{6}=\frac{2}{3}\)

- Read the total number of slices and the number still for sale from the diagram. - Subtract to find the number sold. - Write the sold count over the total and simplify.

1. The image shows \(12\) equal slices, with \(8\) blue-shaded slices remaining. 2. The number sold is \(12-8=4\). 3. The fraction sold is \(\frac{4}{12}=\frac{1}{3}\).

a) \(\frac{1}{3}\) b) \(4\) slices

- Count the total number of tiles shown. - Count each color separately and write each count over the same total. - Simplify both fractions.

1. The image shows \(12\) equal tiles. 2. Four tiles are blue, so the blue fraction is \(\frac{4}{12}=\frac{1}{3}\). 3. Eight tiles are white, so the white fraction is \(\frac{8}{12}=\frac{2}{3}\).

Blue: \(\frac{1}{3}\) White: \(\frac{2}{3}\)

- Count the total number of equal parts in the figure. - Count the tinted parts. - Simplify the resulting fraction with a common factor.

1. The image shows \(6\) equal triangles. 2. Four triangles are tinted, so the tinted fraction is \(\frac{4}{6}\). 3. Divide the numerator and denominator by \(2\): \(\frac{4}{6}=\frac{2}{3}\).

\(\frac{2}{3}\)

- Count all the equal tiles shown. - Count the blue tiles. - Simplify the shaded fraction using a common factor.

1. The image shows \(12\) equal tiles. 2. Nine tiles are blue, so the blue fraction is \(\frac{9}{12}\). 3. Divide the numerator and denominator by \(3\): \(\frac{9}{12}=\frac{3}{4}\).

\(\frac{3}{4}\)

- Count all equal parts or objects for each denominator. - Count the shaded parts or objects for each numerator. - Simplify each fraction by dividing both parts by a common factor.

1. In a), \(3\) of \(6\) equal parts are shaded, so \(\frac{3}{6}=\frac{1}{2}\). 2. In b), \(6\) of \(10\) equal rectangles are shaded, so \(\frac{6}{10}=\frac{3}{5}\). 3. In c), \(4\) of \(12\) objects are shaded, so \(\frac{4}{12}=\frac{1}{3}\).

a) \(\frac{1}{2}\) b) \(\frac{3}{5}\) c) \(\frac{1}{3}\)

- Count all equal sectors in the circle. - Count the blue sectors. - Simplify the fraction using a common factor.

1. The circle has \(10\) equal sectors, and \(4\) are blue. 2. The shaded fraction is \(\frac{4}{10}\). 3. Divide the numerator and denominator by \(2\): \(\frac{4}{10}=\frac{2}{5}\).

The blue-shaded fraction is \(\frac{2}{5}\).

- Count the shaded squares for the numerator. - Use the required denominator to check the total number of equal squares. - Simplify by dividing the numerator and denominator by a common factor.

1. The grid has \(3\times4=12\) equal squares. 2. Nine squares are shaded, so the fraction is \(\frac{9}{12}\). 3. Divide the numerator and denominator by \(3\): \(\frac{9}{12}=\frac{3}{4}\).

\(\frac{9}{12}=\frac{3}{4}\)

- Count all the circles shown. - Count the blue circles. - Write the blue count over the total and simplify.

1. There are \(10\) circles in all. 2. Six circles are blue, so the fraction is \(\frac{6}{10}\). 3. Divide the numerator and denominator by \(2\): \(\frac{6}{10}=\frac{3}{5}\).

\(\frac{3}{5}\)

- Write the number of blue dots over the total number of dots. - The unshaded dots make the rest of the whole group. - Simplify both fractions using a common factor.

1. The blue fraction is \(\frac{6}{10}\). 2. Divide the numerator and denominator by \(2\): \(\frac{6}{10}=\frac{3}{5}\). 3. There are \(10-6=4\) unshaded dots, so the unshaded fraction is \(\frac{4}{10}=\frac{2}{5}\).

Blue: \(\frac{3}{5}\); unshaded: \(\frac{2}{5}\).

- For each group, write the shaded count over the total count. - Simplify each fraction completely. - Compare the two simplified fractions.

1. In group A, \(4\) of \(8\) dots are blue: \(\frac{4}{8}=\frac{1}{2}\). 2. In group B, \(6\) of \(12\) dots are green: \(\frac{6}{12}=\frac{1}{2}\). 3. Since both fractions simplify to \(\frac{1}{2}\), the fractions are equivalent.

In group A, \(\frac{4}{8}=\frac{1}{2}\), and in group B, \(\frac{6}{12}=\frac{1}{2}\). Therefore, the fractions are equivalent.

- Use P's position between \(0\) and \(1\) on line a). - Find the matching location on the fourths line. - Count fourth-sized intervals from \(0\) to that position.

1. On line a), P is at the only tick halfway between \(0\) and \(1\), so P represents \(\frac{1}{2}\). 2. On line b), the same halfway position is two fourth-sized intervals from \(0\). 3. Therefore, the fraction with denominator \(4\) at the same position is \(\frac{2}{4}\).

\(\frac{2}{4}\)

- Pair the eight equal sections two at a time. - Count how many pairs make the whole and how many shaded pairs there are. - Relabel the same shaded amount using those larger equal parts.

1. The bar has \(6\) shaded eighths, so it represents \(\frac{6}{8}\). 2. Pairing the eighths makes \(4\) equal fourth-sized groups in the whole. 3. The \(6\) shaded eighths form \(3\) shaded pairs, so the same amount is \(\frac{3}{4}\). 4. Therefore, \(\frac{6}{8}=\frac{3}{4}\).

\(\frac{3}{4}\)

- Compare each fraction with \(\frac{3}{4}\) one part at a time. - The numerator and denominator must be changed by the same factor. - For part 2), ask what multiplication changes the original denominator into the target denominator.

1. To get denominator \(12\), multiply \(4\) by \(3\). Multiply the numerator by \(3\) too: \(3\times3=9\). 2. To get numerator \(6\), multiply \(3\) by \(2\). Multiply the denominator by \(2\) too: \(4\times2=8\). 3. To change denominator \(4\) to denominator \(8\), multiply by \(2\). The scale factor is \(2\).

1) \(9\) and \(8\) 2) \(2\)

- A number can reduce a fraction only when it divides both the numerator and denominator. - Use divisibility rules for \(2\), \(3\), \(4\), \(6\), and \(9\). - Check each proposed number separately.

1. Both \(126\) and \(162\) are even, so \(2\) works. 2. Each number has digit sum \(9\), so both are divisible by \(3\) and \(9\). 3. Neither number is divisible by \(4\). 4. Both are divisible by \(2\) and \(3\), so both are divisible by \(6\). 5. Therefore, \(2, 3, 6,\) and \(9\) work.

\(2, 3, 6,\) and \(9\)

- Compare each incomplete fraction with the complete fraction at the start of its chain. - Use the same multiplication on the numerator and denominator. - Check that every fraction in a chain names the same amount.

1. In the first chain, \(3\) is multiplied by \(2\) to get \(6\), so \(2\times2=4\). Also, \(2\) is multiplied by \(4\) to get \(8\), so \(3\times4=12\). 2. In the second chain, \(4\) is multiplied by \(2\) to get \(8\), so \(3\times2=6\). Also, \(3\) is multiplied by \(3\) to get \(9\), so \(4\times3=12\).

1) \(4\), \(12\) 2) \(6\), \(12\)

- Compare the original numerator with the simplified numerator to identify the divisor. - The same divisor was used on the denominator. - Work backward from the simplified denominator and then check the original fraction.

1. The original numerator \(9\) became \(3\), so the numerator was divided by \(3\). 2. The denominator must have been divided by the same number. Reverse that step: \(4\times3=12\). 3. The original fraction was \(\frac{9}{12}\), and dividing both parts by \(3\) gives \(\frac{3}{4}\).

The divisor was \(3\), and the original denominator was \(12\).

- Simplify each fraction completely. - Fractions with the same simplest form are equivalent. - Look for the one simplest form that differs from the other three.

1. \(\frac{6}{8}=\frac{3}{4}\). 2. \(\frac{4}{6}=\frac{2}{3}\). 3. \(\frac{9}{12}=\frac{3}{4}\). 4. \(\frac{3}{4}\) is already in simplest form. 5. Therefore, \(\frac{4}{6}\) does not belong because it simplifies to \(\frac{2}{3}\), while the other three equal \(\frac{3}{4}\).

\(\frac{4}{6}\) does not belong.

- An original denominator must divide the target denominator evenly. - List the factor pairs of \(12\). - For part b), choose numerators smaller than their denominators and check that the fractions can scale to twelfths.

1. The positive factors of \(12\) are \(1,2,3,4,6,12\). 2. Excluding \(1\) and \(12\), the possible denominators are \(2,3,4,6\). 3. Examples are \(\frac{1}{2}=\frac{6}{12}\), \(\frac{1}{3}=\frac{4}{12}\), \(\frac{1}{4}=\frac{3}{12}\), and \(\frac{1}{6}=\frac{2}{12}\).

a) \(2,3,4,6\) b) For example, \(\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{6}\), respectively.

- First compare the denominator \(3\) with the denominator \(12\). - Apply the same multiplication to the numerator. - For the last arrow, divide both parts of the middle fraction by the stated number.

1. To change denominator \(3\) to denominator \(12\), multiply by \(4\). 2. Multiply the numerator by the same factor: \(2\times4=8\), so the middle fraction is \(\frac{8}{12}\). 3. Dividing both parts of \(\frac{8}{12}\) by \(2\) gives \(\frac{4}{6}\).

The missing values are \(4\), \(8\), and \(6\), in that order.

- Carry out each student's divisions in the order given. - Compare the combined effect of Leon's two divisions with Sophie's one division. - Think about when a large common factor is easy to spot and when smaller factors may be easier.

1. Leon's method gives \(\frac{8}{12}\rightarrow\frac{4}{6}\rightarrow\frac{2}{3}\). 2. Sophie's method gives \(\frac{8}{12}\rightarrow\frac{2}{3}\) by dividing both parts by \(4\). 3. Leon's two divisors have product \(2\times2=4\), so his repeated divisions have the same effect as Sophie's single division. 4. Sophie's method is faster when a larger common factor is easy to see. Leon's method can be easier when small common factors are noticed first.

a) Both students get \(\frac{2}{3}\). b) Dividing by \(2\) and then by \(2\) has the same effect as dividing by \(4\). c) Sophie's method is faster; Leon's method can be easier when small common factors are easier to notice.

- A common denominator must be a multiple of both original denominators. - Test each proposed number for divisibility by \(6\) and \(10\). - Use the same scale factor for each numerator and denominator.

1. A common denominator must be divisible by both \(6\) and \(10\). 2. The numbers \(30\) and \(60\) are divisible by both. The numbers \(20\) and \(45\) are not. 3. The least common denominator is \(30\). 4. Rewrite the fractions: \(\frac{1}{6}=\frac{5}{30}\) and \(\frac{3}{10}=\frac{9}{30}\).

\(30\) and \(60\) are suitable. Using denominator \(30\), the fractions are \(\frac{5}{30}\) and \(\frac{9}{30}\).

- Find the least number divisible by all three denominators. - Determine the scale factor for each denominator. - Multiply each numerator by the same factor used for its denominator.

1. The least common multiple of \(10\), \(15\), and \(6\) is \(30\). 2. \(\frac{3}{10}=\frac{3\times3}{10\times3}=\frac{9}{30}\). 3. \(\frac{4}{15}=\frac{4\times2}{15\times2}=\frac{8}{30}\). 4. \(\frac{1}{6}=\frac{1\times5}{6\times5}=\frac{5}{30}\).

\(\frac{9}{30},\frac{8}{30},\frac{5}{30}\)

- Find the least common multiple of the denominators in each set. - Determine the scale factor for each original denominator. - Multiply each numerator and denominator by the same factor.

1. For a), the least common denominator is \(24\): \(\frac{5}{6}=\frac{20}{24}\) and \(\frac{7}{8}=\frac{21}{24}\). 2. For b), the least common denominator is \(36\): \(\frac{2}{9}=\frac{8}{36}\) and \(\frac{5}{12}=\frac{15}{36}\). 3. For c), the least common denominator is \(12\): \(\frac{3}{4}=\frac{9}{12}\), \(\frac{1}{6}=\frac{2}{12}\), and \(\frac{2}{3}=\frac{8}{12}\).

a) \(\frac{20}{24}\) and \(\frac{21}{24}\) b) \(\frac{8}{36}\) and \(\frac{15}{36}\) c) \(\frac{9}{12},\frac{2}{12},\frac{8}{12}\)

- A counterexample is one example that makes a general claim false. - Look for two odd numbers that share a common factor greater than \(1\). - Check that your fraction is not equal to \(1\), then simplify it.

1. One valid choice is \(\frac{9}{3}\). 2. Both \(9\) and \(3\) are odd, and they have the common factor \(3\). 3. Divide numerator and denominator by \(3\): \(\frac{9}{3}=3\). 4. This fraction satisfies Jan's conditions but still simplifies, so it is a counterexample.

One possible answer is \(\frac{9}{3}=3\). Both numerator and denominator are odd, and both are divisible by \(3\), so Jan's claim is false.

- Multiply both coordinates by each scale factor. - Compare consecutive x-coordinates and consecutive y-coordinates. - Look for a constant horizontal and vertical change.

1. Multiplying by \(2\) gives \(\frac{4}{6}\), so the point is \((6, 4)\). 2. Multiplying by \(3\) gives \(\frac{6}{9}\), so the point is \((9, 6)\). 3. Multiplying by \(4\) gives \(\frac{8}{12}\), so the point is \((12, 8)\). 4. From one point to the next, add \((3, 2)\): move \(3\) units right and \(2\) units up.

a) \((6, 4)\), \((9, 6)\), and \((12, 8)\) b) Move \(3\) units right and \(2\) units up each time.

- How many squares are one-fourth of \(16\)? - How many squares are one-eighth of \(16\)? - After accounting for the tulips and daffodils, how many squares remain? - Write the rose squares as a fraction of all \(16\) squares.

1. Find the tulip squares: \(\frac{1}{4} \times 16 = 4\). 2. Find the daffodil squares: \(\frac{1}{8} \times 16 = 2\). 3. Find the number of planted squares already used: \(4 + 2 = 6\). 4. Find the rose squares: \(16 - 6 = 10\). 5. The rose fraction is \(\frac{10}{16}\), which simplifies to \(\frac{5}{8}\).

There are \(10\) squares planted with roses. The roses cover \(\frac{5}{8}\) of the bed (or \(\frac{10}{16}\)).

- Write the shaded count over the total number of equal parts for each figure. - Simplify each fraction before comparing them. - Look for two figures whose simplest forms match.

1. In a), \(4\) of \(6\) equal sectors are shaded: \(\frac{4}{6}=\frac{2}{3}\). 2. In b), \(9\) of \(12\) equal squares are shaded: \(\frac{9}{12}=\frac{3}{4}\). 3. In c), \(4\) of \(6\) equal parts are shaded: \(\frac{4}{6}=\frac{2}{3}\). 4. In d), \(2\) of \(4\) equal parts are shaded: \(\frac{2}{4}=\frac{1}{2}\). 5. Figures a) and c) both represent \(\frac{2}{3}\).

a) \(\frac{2}{3}\) b) \(\frac{3}{4}\) c) \(\frac{2}{3}\) d) \(\frac{1}{2}\) Figures a) and c) match.

- For each figure, count the shaded parts and the total equal parts. - Simplify each fraction by dividing the numerator and denominator by a common factor. - Compare the simplified fractions.

1. In figure a), \(6\) of the \(8\) equal parts are shaded, so the fraction is \(\frac{6}{8}=\frac{3}{4}\). 2. In figure b), the grid has \(3\times4=12\) equal squares. Nine are shaded, so the fraction is \(\frac{9}{12}=\frac{3}{4}\). 3. Both fractions simplify to \(\frac{3}{4}\), so they are equivalent and represent the same portion of a whole.

a) \(\frac{6}{8}=\frac{3}{4}\) b) \(\frac{9}{12}=\frac{3}{4}\) Both figures show the same fraction of a whole.

- Count the equal parts and the blue parts in each figure. - Simplify each fraction. - Compare the simplified fractions rather than the number of dividing lines.

1. Figure 1 has \(1\) of \(4\) equal parts shaded, so it represents \(\frac{1}{4}\). 2. Figure 2 has \(2\) of \(8\) equal parts shaded, so it represents \(\frac{2}{8}\). 3. Figure 3 has \(2\) of \(6\) equal parts shaded, so it represents \(\frac{2}{6}\). 4. Simplify \(\frac{2}{8}\) by dividing the numerator and denominator by \(2\): \(\frac{2}{8} = \frac{1}{4}\). Therefore, figures 1 and 2 show the same fraction. Figure 3 represents \(\frac{2}{6} = \frac{1}{3}\).

a) Figure 1: \(\frac{1}{4}\); Figure 2: \(\frac{2}{8}\); Figure 3: \(\frac{2}{6}\) b) Figures 1 and 2 show the same fraction because \(\frac{2}{8} = \frac{1}{4}\).

- A unit fraction has numerator \(1\). - Simplify each candidate only as far as needed to decide whether its numerator can become \(1\). - Look for a common factor shared by the numerator and \(12\).

1. A unit fraction has numerator \(1\), so each fraction must simplify to a fraction with numerator \(1\). 2. \(\frac{1}{12}\) is already a unit fraction. Also, \(\frac{2}{12}=\frac{1}{6}\), \(\frac{3}{12}=\frac{1}{4}\), \(\frac{4}{12}=\frac{1}{3}\), and \(\frac{6}{12}=\frac{1}{2}\). 3. The remaining fractions from \(\frac{5}{12}\) through \(\frac{11}{12}\) do not simplify to a numerator of \(1\).

\(\frac{1}{12}=\frac{1}{12}\), \(\frac{2}{12}=\frac{1}{6}\), \(\frac{3}{12}=\frac{1}{4}\), \(\frac{4}{12}=\frac{1}{3}\), \(\frac{6}{12}=\frac{1}{2}\)