(a) Simplify, without using tables or calculator : (frac{frac{3}{4}(3frac{3}{8} + 1frac{5}{8})}{2frac{1}{8} – 1frac{1}{2}}).
(b) Given that (log_{10} 2 = 0.3010) and (log_{10} 3 = 0.4771), evaluate, correct to 2 significant figures and without using tables or calculator, (log_{10} 1.125).
Explanation
(a) (frac{frac{3}{4}(3frac{3}{8} + 1frac{5}{8})}{2frac{1}{8} – 1frac{1}{2}})
(frac{3}{4}(3frac{3}{8} + 1frac{5}{8}) = frac{3}{4}(frac{27}{8} + frac{13}{8}))
= (frac{3}{4}(frac{40}{8}))
= (frac{15}{4})
(2frac{1}{8} – 1frac{1}{2} = frac{17}{8} – frac{3}{2})
= (frac{17}{8} – frac{12}{8})
= (frac{5}{8})
(therefore frac{frac{3}{4}(3frac{3}{8} + 1frac{5}{8})}{2frac{1}{8} – 1frac{1}{2}} = frac{15}{4} div frac{5}{8})
= (frac{15}{4} times frac{8}{5})
= (6)
(b) (log_{10} 2 = 0.3010 ; log_{10} 3 = 0.4771)
(log_{10} 1.125 = log_{10}(frac{1125}{1000})) (Dividing through with 125)
= (log_{10} (frac{9}{8}))
= (log_{10} 9 – log_{10} 8)
(log_{10} 9 = log_{10} 3^{2} = 2log_{10} 3 = 2 times 0.4771 = 0.9542)
(log_{10} 8 = log_{10} 2^{3} = 3log_{10} 2 = 3 times 0.3010 = 0.9030)
(therefore log_{10} (frac{9}{8}) = 0.9542 – 0.9030 = 0.0512)
(approxeq 0.051) ( 2 significant figures)