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Mathematics Question on Need For Rational Numbers

Write four more rational numbers in each of the following patterns:
(i) −35,−610,−915,−1220\frac{-3}{5},\frac{-6}{10},\frac{-9}{15},\frac{-12}{20}...

(ii) −14,−28,−312,\frac{-1}{4},\frac{-2}{8},\frac{-3}{12},...

(iii) −16,2−12,3−18,4−24\frac{-1}{6},\frac{2}{-12},\frac{3}{-18},\frac{4}{-24},....

(iv)−23,2−3,4−6,6−9\frac{-2}{3},\frac{2}{-3},\frac{4}{-6},\frac{6}{-9},...

Answer

(i)−35,−610,−915,−1220\frac{-3}{5},\frac{-6}{10},\frac{-9}{15},\frac{-12}{20},...

= −35,−3×25×2,−3×35×3,−3×45×4\frac{-3}{5},\frac{-3\times2}{5\times2},\frac{-3\times3}{5\times3},\frac{-3\times4}{5\times4},...

It can be observed that the numerator is a multiple of 3 while the denominator is a multiple of 5 and as we increase them further, these multiples are increasing. Therefore, the next four rational numbers in this pattern are
= −3×55×5,−3×65×6,−3×75×7,−3×85×8\frac{-3\times5}{5\times5},\frac{-3\times6}{5\times6},\frac{-3\times7}{5\times7},\frac{-3\times8}{5\times8}...

= −1525,−1830,−2135,−2440\frac{-15}{25},\frac{-18}{30},\frac{-21}{35},\frac{-24}{40}...

(ii) −14,−28,−312,\frac{-1}{4},\frac{-2}{8},\frac{-3}{12},...

= −14,−1×24×2,−1×34×3.\frac{-1}{4},\frac{-1\times2}{4\times2},\frac{-1\times3}{4\times3}...

The next four rational numbers in this pattern are.
=−1×44×4,−1×54×5,−1×64×6,−1×74×7.\frac{-1\times4}{4\times4},\frac{-1\times5}{4\times5},\frac{-1\times6}{4\times6},\frac{-1\times7}{4\times7}....

=−416,−520,−624,−728...= \frac{-4}{16},\frac{-5}{20},\frac{-6}{24},\frac{-7}{28}...

(iii)−16,2−12,3−18,4−24,....\frac{-1}{6},\frac{2}{-12},\frac{3}{-18},\frac{4}{-24},....

=−16,1×2−6×2,1×3−6×3,1×4−6×4...= \frac{-1}{6},\frac{1\times2}{-6\times2},\frac{1\times3}{-6\times3},\frac{1\times4}{-6\times4}...

The next four rational numbers in this pattern are
1×5−6×5,1×6−6×6,1×7−6×7,1×8−6×8....\frac{1\times5}{-6\times5},\frac{1\times6}{-6\times6},\frac{1\times7}{-6\times7},\frac{1\times8}{-6\times8}....

5−30,6−36,7−42,8−48...\frac{5}{-30},\frac{6}{-36},\frac{7}{-42},\frac{8}{-48}...

(iv) −23,2−3,4−6,6−9,...\frac{-2}{3},\frac{2}{-3},\frac{4}{-6},\frac{6}{-9},...

−23,2−3,2×2−3×2,2×3−3×3...\frac{-2}{3},\frac{2}{-3},\frac{2\times2}{-3\times2},\frac{2\times3}{-3\times3}...

The next four rational numbers in this pattern are
2×4−3×4,2×5−3×5,2×6−3×6,2×7−3×7...\frac{2\times4}{-3\times4},\frac{2\times5}{-3\times5},\frac{2\times6}{-3\times6},\frac{2\times7}{-3\times7}...

8−12,10−15,12−18,14−21....\frac{8}{-12},\frac{10}{-15},\frac{12}{-18},\frac{14}{-21}....