Lesson 4 · from §71 of the 1910 text
Equations with Decimals
Equations with decimals
Decimals are fractions with denominators 10, 100, 1000, and so on. Either work with the decimals directly, or clear them by multiplying both members by the right power of 10.
Example. Solve .4x + .7 = 9.7 - .05x.
Transposing, .4x + .05x = 9.7 - .7, so .45x = 9 and x = \frac{9}{.45} = 20. Or multiply everything by 100 first: 40x + 70 = 970 - 5x, so 45x = 900 and x = 20.
When decimals sit inside fractions, clear the ordinary denominators first, then the decimals: \frac{.32x}{.05} = 6.4x, \qquad \frac{.045x}{.125} = .36x.
Try one: solve .3(x - 2) = .05x + .4.
Quick check
\displaystyle .3(x-2)=.05x+.4
From the history of algebra
Our numerals 1, 2, 3, … came from India, where a place-value system with a zero was in use by about 600 AD. Arab scholars, al-Khwarizmi among them, passed it west, and Leonardo of Pisa (Fibonacci) pushed for it in Europe in 1202. Decimal fractions came much later: Simon Stevin's little book De Thiende ("The Tenth," 1585) showed merchants how to calculate with them, though his notation was clumsy. The decimal point itself was made popular by John Napier, the inventor of logarithms, around 1617.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Solve. Decimal answers like 1.42 are fine.
\displaystyle .3x+4=.25
\displaystyle .15x-.4x=235x-2352.5
\displaystyle 1.3x+8.24=-5.26-3.2x
\displaystyle 3x-1.245x+.6x=1.5+.355x
\displaystyle 3.5x+.0564-.1x=4.9128-.02x
\displaystyle .12(2x+.05)-.15(1.5x-2)=0.246
\displaystyle \frac{.01x+.003}{6}+\frac{.02x+.0008}{7}=.0017
\displaystyle \frac{.3(x+5)}{8}-\frac{4(.25x-.35)}{7}=\frac{14.325}{56}
\displaystyle \frac{0.5(6-.2x)}{.80}-\frac{.3(.4x-3)}{.16}=5
\displaystyle \frac{.32x}{.05}+\frac{.045x}{.125}=13.52
\displaystyle \frac{33}{x+5}+\frac{3.75}{.5(x-8.5)}=0