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Question

# Verify commutativity of addition of rational numbers for each of the following pairs of rational numbers: (i) −115 and 47 (ii)49 and 7−12 (iii) −35 and −2−15 (iv) 2−7 and 12−35 (v) 4 and −35 (vi) −4 and 4−7

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Solution

## (i) −115 and 47−115+47∴−115=−11×75×7=−7735and 47=4×57×5=2035∴−115+47=−7735+2035=−77+2035=−5735and47+−115=2035−7735=20−7735=−5735∴−115+47=47+−115 (ii) 49 and 7−127×(−1)−12×(−1)=−712Now 49+−712LCM of 9,12=36∴49=4×49×4=1636−712=−7×312×3=−2136∴49+−712+49 (iii) −35 and −2−15−2−15=−2×(−1)−15×(−1)=215LCM of 5 and 15=15∴−35=−3×35×3=−915Now −35+215=−915+215−9+215=−715and 215+−35=215+−915=2−915=−715∴−35+215=215+−35 (iv) 2−7 and 12−35LCM of 7,35=352−7=2×(−5)−7×(−5)=−103512−35=12×(−1)−35×(−1)=−1235Now −27+−1235=−1035+−1235=−10−1235=−2235and −1235+−27=−1235+−1035=−12−1035=−2235∴−27+−1235=−1235+−27 (v) 4 and −35LCM of 1,5=5∴41=4×51×5=205Now 41+−35=205+−35=20−35=175and −35+41=−35+205=−3+205=175∴4+−35=−35+4 (vi) −4 and 4−7−41 and 4−7(∵4−7=4×(−1)−7×(−1)=−47)∴−41=−4×71×7=−287and−47=−28−47=−327and −47+(−4)=−47+−287=−4−287=−327∴−4+−47=−47+(−4)

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