Answer :
Let's fill in the blanks to make each pair of fractions equivalent.
1. [tex]\(\frac{3}{7} = \frac{9}{\square}\)[/tex]
To make the fractions equivalent, we need to solve for the missing number:
[tex]\[ \frac{3}{7} = \frac{9}{x} \][/tex]
Cross-multiply:
[tex]\[ 3x = 9 \cdot 7 \][/tex]
[tex]\[ 3x = 63 \][/tex]
Divide both sides by 3:
[tex]\[ x = 21 \][/tex]
Thus, [tex]\(\frac{3}{7} = \frac{9}{21}\)[/tex].
2. [tex]\(\frac{8}{12} = \frac{2}{\square}\)[/tex]
To make the fractions equivalent, we need to solve for the missing number:
[tex]\[ \frac{8}{12} = \frac{2}{y} \][/tex]
Simplify [tex]\(\frac{8}{12}\)[/tex] first:
[tex]\[ \frac{8}{12} = \frac{2}{3} \][/tex]
Now we have:
[tex]\[ \frac{2}{3} = \frac{2}{y} \][/tex]
Therefore, [tex]\(y = 3\)[/tex].
3. [tex]\(\frac{\pi}{10} = \frac{1}{5}\)[/tex]
This statement is incorrect. The fractions are not equivalent, so we skip this.
4. [tex]\(\frac{10}{40} = \frac{2}{\square}\)[/tex]
To make the fractions equivalent, we need to solve for the missing number:
[tex]\[ \frac{10}{40} = \frac{2}{z} \][/tex]
Simplify [tex]\(\frac{10}{40}\)[/tex]:
[tex]\[ \frac{10}{40} = \frac{1}{4} \][/tex]
Now we have:
[tex]\[ \frac{1}{4} = \frac{2}{z} \][/tex]
Cross-multiply:
[tex]\[ 1z = 2 \cdot 4 \][/tex]
[tex]\[ z = 8 \][/tex]
Thus, [tex]\(\frac{10}{40} = \frac{2}{8}\)[/tex].
5. [tex]\(\frac{30}{50} = \frac{1}{\square}\)[/tex]
To make the fractions equivalent, we need to solve for the missing number:
[tex]\[ \frac{30}{50} = \frac{1}{k} \][/tex]
Simplify [tex]\(\frac{30}{50}\)[/tex]:
[tex]\[ \frac{30}{50} = \frac{3}{5} \][/tex]
Now we have:
[tex]\[ \frac{3}{5} = \frac{1}{k} \][/tex]
Cross-multiply:
[tex]\[ 3k = 5 \][/tex]
Divide both sides by 3:
[tex]\[ k = \frac{5}{3} \][/tex]
Thus, [tex]\(\frac{30}{50} = \frac{1}{\frac{5}{3}}\)[/tex].
6. [tex]\(\frac{2}{10} < \frac{1}{59}\)[/tex]
This inequality is false. It is true that [tex]\(\frac{2}{10} > \frac{1}{59}\)[/tex].
7. [tex]\(\frac{\square}{6} = \frac{3}{18}\)[/tex]
To make the fractions equivalent, we need to solve for the missing number:
[tex]\[ \frac{a}{6} = \frac{3}{18} \][/tex]
Simplify [tex]\(\frac{3}{18}\)[/tex]:
[tex]\[ \frac{3}{18} = \frac{1}{6} \][/tex]
Now we have:
[tex]\[ \frac{a}{6} = \frac{1}{6} \][/tex]
Therefore, [tex]\(a = 1\)[/tex].
8. [tex]\(\frac{24}{\sqrt{7}} = \frac{4}{5}\)[/tex]
To make the fractions equivalent, we need to solve for the missing number:
Cross-multiplying:
[tex]\[ 24 \cdot 5 = 4 \cdot \sqrt{7} \][/tex]
[tex]\[ 120 = 4 \cdot \sqrt{7} \][/tex]
Divide both sides by 4:
[tex]\[ 30 = \sqrt{7} \][/tex]
Thus, [tex]\(\sqrt{7} \approx 2.6457513110645907\)[/tex].
9. [tex]\(\frac{6}{\square} = \frac{4}{5}\)[/tex]
To make the fractions equivalent, we need to solve for the missing number:
[tex]\[ \frac{6}{p} = \frac{4}{5} \][/tex]
Cross-multiply:
[tex]\[ 6 \cdot 5 = 4 \cdot p \][/tex]
[tex]\[ 30 = 4p \][/tex]
Divide both sides by 4:
[tex]\[ p = \frac{30}{4} \][/tex]
[tex]\[ p = 7.5 \][/tex]
Thus, [tex]\(\frac{6}{7.5} = \frac{4}{5}\)[/tex].
In summary:
1. [tex]\(21\)[/tex]
2. [tex]\(3\)[/tex]
4. [tex]\(8\)[/tex]
5. [tex]\(\frac{5}{3}\)[/tex]
7. [tex]\(1\)[/tex]
8. [tex]\(2.6457513110645907\)[/tex]
9. [tex]\(7.5\)[/tex]
1. [tex]\(\frac{3}{7} = \frac{9}{\square}\)[/tex]
To make the fractions equivalent, we need to solve for the missing number:
[tex]\[ \frac{3}{7} = \frac{9}{x} \][/tex]
Cross-multiply:
[tex]\[ 3x = 9 \cdot 7 \][/tex]
[tex]\[ 3x = 63 \][/tex]
Divide both sides by 3:
[tex]\[ x = 21 \][/tex]
Thus, [tex]\(\frac{3}{7} = \frac{9}{21}\)[/tex].
2. [tex]\(\frac{8}{12} = \frac{2}{\square}\)[/tex]
To make the fractions equivalent, we need to solve for the missing number:
[tex]\[ \frac{8}{12} = \frac{2}{y} \][/tex]
Simplify [tex]\(\frac{8}{12}\)[/tex] first:
[tex]\[ \frac{8}{12} = \frac{2}{3} \][/tex]
Now we have:
[tex]\[ \frac{2}{3} = \frac{2}{y} \][/tex]
Therefore, [tex]\(y = 3\)[/tex].
3. [tex]\(\frac{\pi}{10} = \frac{1}{5}\)[/tex]
This statement is incorrect. The fractions are not equivalent, so we skip this.
4. [tex]\(\frac{10}{40} = \frac{2}{\square}\)[/tex]
To make the fractions equivalent, we need to solve for the missing number:
[tex]\[ \frac{10}{40} = \frac{2}{z} \][/tex]
Simplify [tex]\(\frac{10}{40}\)[/tex]:
[tex]\[ \frac{10}{40} = \frac{1}{4} \][/tex]
Now we have:
[tex]\[ \frac{1}{4} = \frac{2}{z} \][/tex]
Cross-multiply:
[tex]\[ 1z = 2 \cdot 4 \][/tex]
[tex]\[ z = 8 \][/tex]
Thus, [tex]\(\frac{10}{40} = \frac{2}{8}\)[/tex].
5. [tex]\(\frac{30}{50} = \frac{1}{\square}\)[/tex]
To make the fractions equivalent, we need to solve for the missing number:
[tex]\[ \frac{30}{50} = \frac{1}{k} \][/tex]
Simplify [tex]\(\frac{30}{50}\)[/tex]:
[tex]\[ \frac{30}{50} = \frac{3}{5} \][/tex]
Now we have:
[tex]\[ \frac{3}{5} = \frac{1}{k} \][/tex]
Cross-multiply:
[tex]\[ 3k = 5 \][/tex]
Divide both sides by 3:
[tex]\[ k = \frac{5}{3} \][/tex]
Thus, [tex]\(\frac{30}{50} = \frac{1}{\frac{5}{3}}\)[/tex].
6. [tex]\(\frac{2}{10} < \frac{1}{59}\)[/tex]
This inequality is false. It is true that [tex]\(\frac{2}{10} > \frac{1}{59}\)[/tex].
7. [tex]\(\frac{\square}{6} = \frac{3}{18}\)[/tex]
To make the fractions equivalent, we need to solve for the missing number:
[tex]\[ \frac{a}{6} = \frac{3}{18} \][/tex]
Simplify [tex]\(\frac{3}{18}\)[/tex]:
[tex]\[ \frac{3}{18} = \frac{1}{6} \][/tex]
Now we have:
[tex]\[ \frac{a}{6} = \frac{1}{6} \][/tex]
Therefore, [tex]\(a = 1\)[/tex].
8. [tex]\(\frac{24}{\sqrt{7}} = \frac{4}{5}\)[/tex]
To make the fractions equivalent, we need to solve for the missing number:
Cross-multiplying:
[tex]\[ 24 \cdot 5 = 4 \cdot \sqrt{7} \][/tex]
[tex]\[ 120 = 4 \cdot \sqrt{7} \][/tex]
Divide both sides by 4:
[tex]\[ 30 = \sqrt{7} \][/tex]
Thus, [tex]\(\sqrt{7} \approx 2.6457513110645907\)[/tex].
9. [tex]\(\frac{6}{\square} = \frac{4}{5}\)[/tex]
To make the fractions equivalent, we need to solve for the missing number:
[tex]\[ \frac{6}{p} = \frac{4}{5} \][/tex]
Cross-multiply:
[tex]\[ 6 \cdot 5 = 4 \cdot p \][/tex]
[tex]\[ 30 = 4p \][/tex]
Divide both sides by 4:
[tex]\[ p = \frac{30}{4} \][/tex]
[tex]\[ p = 7.5 \][/tex]
Thus, [tex]\(\frac{6}{7.5} = \frac{4}{5}\)[/tex].
In summary:
1. [tex]\(21\)[/tex]
2. [tex]\(3\)[/tex]
4. [tex]\(8\)[/tex]
5. [tex]\(\frac{5}{3}\)[/tex]
7. [tex]\(1\)[/tex]
8. [tex]\(2.6457513110645907\)[/tex]
9. [tex]\(7.5\)[/tex]
