Answer :
To find five rational numbers equivalent to the given rational numbers, we multiply both the numerator and denominator of each fraction by the same integer. Specifically, let's use the integers 2, 3, 4, 5, and 6 for this purpose. Here's the step-by-step solution:
### (i) [tex]\(\frac{3}{5}\)[/tex]
1. Multiply by 2:
[tex]\[ \frac{3 \times 2}{5 \times 2} = \frac{6}{10} \][/tex]
2. Multiply by 3:
[tex]\[ \frac{3 \times 3}{5 \times 3} = \frac{9}{15} \][/tex]
3. Multiply by 4:
[tex]\[ \frac{3 \times 4}{5 \times 4} = \frac{12}{20} \][/tex]
4. Multiply by 5:
[tex]\[ \frac{3 \times 5}{5 \times 5} = \frac{15}{25} \][/tex]
5. Multiply by 6:
[tex]\[ \frac{3 \times 6}{5 \times 6} = \frac{18}{30} \][/tex]
Equivalent rational numbers to [tex]\(\frac{3}{5}\)[/tex] are:
[tex]\[ \frac{6}{10}, \frac{9}{15}, \frac{12}{20}, \frac{15}{25}, \frac{18}{30} \][/tex]
### (ii) [tex]\(\frac{-6}{11}\)[/tex]
1. Multiply by 2:
[tex]\[ \frac{-6 \times 2}{11 \times 2} = \frac{-12}{22} \][/tex]
2. Multiply by 3:
[tex]\[ \frac{-6 \times 3}{11 \times 3} = \frac{-18}{33} \][/tex]
3. Multiply by 4:
[tex]\[ \frac{-6 \times 4}{11 \times 4} = \frac{-24}{44} \][/tex]
4. Multiply by 5:
[tex]\[ \frac{-6 \times 5}{11 \times 5} = \frac{-30}{55} \][/tex]
5. Multiply by 6:
[tex]\[ \frac{-6 \times 6}{11 \times 6} = \frac{-36}{66} \][/tex]
Equivalent rational numbers to [tex]\(\frac{-6}{11}\)[/tex] are:
[tex]\[ \frac{-12}{22}, \frac{-18}{33}, \frac{-24}{44}, \frac{-30}{55}, \frac{-36}{66} \][/tex]
### (iii) [tex]\(\frac{7}{-10}\)[/tex]
1. Multiply by 2:
[tex]\[ \frac{7 \times 2}{-10 \times 2} = \frac{14}{-20} \][/tex]
2. Multiply by 3:
[tex]\[ \frac{7 \times 3}{-10 \times 3} = \frac{21}{-30} \][/tex]
3. Multiply by 4:
[tex]\[ \frac{7 \times 4}{-10 \times 4} = \frac{28}{-40} \][/tex]
4. Multiply by 5:
[tex]\[ \frac{7 \times 5}{-10 \times 5} = \frac{35}{-50} \][/tex]
5. Multiply by 6:
[tex]\[ \frac{7 \times 6}{-10 \times 6} = \frac{42}{-60} \][/tex]
Equivalent rational numbers to [tex]\(\frac{7}{-10}\)[/tex] are:
[tex]\[ \frac{14}{-20}, \frac{21}{-30}, \frac{28}{-40}, \frac{35}{-50}, \frac{42}{-60} \][/tex]
### (iv) [tex]\(\frac{8}{15}\)[/tex]
1. Multiply by 2:
[tex]\[ \frac{8 \times 2}{15 \times 2} = \frac{16}{30} \][/tex]
2. Multiply by 3:
[tex]\[ \frac{8 \times 3}{15 \times 3} = \frac{24}{45} \][/tex]
3. Multiply by 4:
[tex]\[ \frac{8 \times 4}{15 \times 4} = \frac{32}{60} \][/tex]
4. Multiply by 5:
[tex]\[ \frac{8 \times 5}{15 \times 5} = \frac{40}{75} \][/tex]
5. Multiply by 6:
[tex]\[ \frac{8 \times 6}{15 \times 6} = \frac{48}{90} \][/tex]
Equivalent rational numbers to [tex]\(\frac{8}{15}\)[/tex] are:
[tex]\[ \frac{16}{30}, \frac{24}{45}, \frac{32}{60}, \frac{40}{75}, \frac{48}{90} \][/tex]
So, the five rational numbers equivalent to each given rational number are:
1. For [tex]\(\frac{3}{5}\)[/tex]: [tex]\(\frac{6}{10}, \frac{9}{15}, \frac{12}{20}, \frac{15}{25}, \frac{18}{30}\)[/tex]
2. For [tex]\(\frac{-6}{11}\)[/tex]: [tex]\(\frac{-12}{22}, \frac{-18}{33}, \frac{-24}{44}, \frac{-30}{55}, \frac{-36}{66}\)[/tex]
3. For [tex]\(\frac{7}{-10}\)[/tex]: [tex]\(\frac{14}{-20}, \frac{21}{-30}, \frac{28}{-40}, \frac{35}{-50}, \frac{42}{-60}\)[/tex]
4. For [tex]\(\frac{8}{15}\)[/tex]: [tex]\(\frac{16}{30}, \frac{24}{45}, \frac{32}{60}, \frac{40}{75}, \frac{48}{90}\)[/tex]
### (i) [tex]\(\frac{3}{5}\)[/tex]
1. Multiply by 2:
[tex]\[ \frac{3 \times 2}{5 \times 2} = \frac{6}{10} \][/tex]
2. Multiply by 3:
[tex]\[ \frac{3 \times 3}{5 \times 3} = \frac{9}{15} \][/tex]
3. Multiply by 4:
[tex]\[ \frac{3 \times 4}{5 \times 4} = \frac{12}{20} \][/tex]
4. Multiply by 5:
[tex]\[ \frac{3 \times 5}{5 \times 5} = \frac{15}{25} \][/tex]
5. Multiply by 6:
[tex]\[ \frac{3 \times 6}{5 \times 6} = \frac{18}{30} \][/tex]
Equivalent rational numbers to [tex]\(\frac{3}{5}\)[/tex] are:
[tex]\[ \frac{6}{10}, \frac{9}{15}, \frac{12}{20}, \frac{15}{25}, \frac{18}{30} \][/tex]
### (ii) [tex]\(\frac{-6}{11}\)[/tex]
1. Multiply by 2:
[tex]\[ \frac{-6 \times 2}{11 \times 2} = \frac{-12}{22} \][/tex]
2. Multiply by 3:
[tex]\[ \frac{-6 \times 3}{11 \times 3} = \frac{-18}{33} \][/tex]
3. Multiply by 4:
[tex]\[ \frac{-6 \times 4}{11 \times 4} = \frac{-24}{44} \][/tex]
4. Multiply by 5:
[tex]\[ \frac{-6 \times 5}{11 \times 5} = \frac{-30}{55} \][/tex]
5. Multiply by 6:
[tex]\[ \frac{-6 \times 6}{11 \times 6} = \frac{-36}{66} \][/tex]
Equivalent rational numbers to [tex]\(\frac{-6}{11}\)[/tex] are:
[tex]\[ \frac{-12}{22}, \frac{-18}{33}, \frac{-24}{44}, \frac{-30}{55}, \frac{-36}{66} \][/tex]
### (iii) [tex]\(\frac{7}{-10}\)[/tex]
1. Multiply by 2:
[tex]\[ \frac{7 \times 2}{-10 \times 2} = \frac{14}{-20} \][/tex]
2. Multiply by 3:
[tex]\[ \frac{7 \times 3}{-10 \times 3} = \frac{21}{-30} \][/tex]
3. Multiply by 4:
[tex]\[ \frac{7 \times 4}{-10 \times 4} = \frac{28}{-40} \][/tex]
4. Multiply by 5:
[tex]\[ \frac{7 \times 5}{-10 \times 5} = \frac{35}{-50} \][/tex]
5. Multiply by 6:
[tex]\[ \frac{7 \times 6}{-10 \times 6} = \frac{42}{-60} \][/tex]
Equivalent rational numbers to [tex]\(\frac{7}{-10}\)[/tex] are:
[tex]\[ \frac{14}{-20}, \frac{21}{-30}, \frac{28}{-40}, \frac{35}{-50}, \frac{42}{-60} \][/tex]
### (iv) [tex]\(\frac{8}{15}\)[/tex]
1. Multiply by 2:
[tex]\[ \frac{8 \times 2}{15 \times 2} = \frac{16}{30} \][/tex]
2. Multiply by 3:
[tex]\[ \frac{8 \times 3}{15 \times 3} = \frac{24}{45} \][/tex]
3. Multiply by 4:
[tex]\[ \frac{8 \times 4}{15 \times 4} = \frac{32}{60} \][/tex]
4. Multiply by 5:
[tex]\[ \frac{8 \times 5}{15 \times 5} = \frac{40}{75} \][/tex]
5. Multiply by 6:
[tex]\[ \frac{8 \times 6}{15 \times 6} = \frac{48}{90} \][/tex]
Equivalent rational numbers to [tex]\(\frac{8}{15}\)[/tex] are:
[tex]\[ \frac{16}{30}, \frac{24}{45}, \frac{32}{60}, \frac{40}{75}, \frac{48}{90} \][/tex]
So, the five rational numbers equivalent to each given rational number are:
1. For [tex]\(\frac{3}{5}\)[/tex]: [tex]\(\frac{6}{10}, \frac{9}{15}, \frac{12}{20}, \frac{15}{25}, \frac{18}{30}\)[/tex]
2. For [tex]\(\frac{-6}{11}\)[/tex]: [tex]\(\frac{-12}{22}, \frac{-18}{33}, \frac{-24}{44}, \frac{-30}{55}, \frac{-36}{66}\)[/tex]
3. For [tex]\(\frac{7}{-10}\)[/tex]: [tex]\(\frac{14}{-20}, \frac{21}{-30}, \frac{28}{-40}, \frac{35}{-50}, \frac{42}{-60}\)[/tex]
4. For [tex]\(\frac{8}{15}\)[/tex]: [tex]\(\frac{16}{30}, \frac{24}{45}, \frac{32}{60}, \frac{40}{75}, \frac{48}{90}\)[/tex]
