GE142 - Course Introduction

3. Decimals

Dr Robert Batzinger
Instructor Emeritus

8/15/22

1 Chapter 3: Decimals

1.1 To read or write decimals

\[\eqalign{1.000000& & 10^0& &\hbox{units}\\ 0.100000& &10^{-1}& &\hbox{tenths}\\ 0.010000& &10^{-2}& &\hbox{hundredths}\\ 0.001000& &10^{-3}& &\hbox{thousandths}\\ 0.000100& &10^{-4}& &\hbox{ten-thousandths}\\ 0.000010& &10^{-5}& &\hbox{hundred-thousandths}\\ 0.00000& &10^{-6}& &\hbox{millionths}\\ }\]

1.2 Powers of ten

\[\eqalign{10^0 & & units& &| & &10^{-6}& & micro\\ 10^{-1}& & deci& &| & &10^{-9}& & nano\\ 10^{-2}& & centi& &| & &10^{-12}& & pico\\ 10^{-3}& & milli& & |& &10^{-15}& & femto\\ }\]

1.3 Naming decimals

\[0.213 = \frac{2}{10} + \frac{1}{100} +\frac{3}{1000}\] Two hundred thirteen thousandths

1.4 Femtosecond photography

  • https://youtu.be/7Z8EtlBe8Ts
  • https://youtu.be/VgjyPmFKxCU
  • https://youtu.be/7Ys_yKGNFRQ

1.5 To find fraction equivalent to a decimal

\[\eqalign{ \frac{1}{7} &=& \frac{x}{1000}\\ \color{yellow}{\frac{1}{7}\times} 1000 &=& 7x\color{yellow}{\times\frac{1}{7}}\\ 0.143 \approx \frac{142.857}{1000} &=& x\\ }\]

1.6 To compare decimals

  • Read an integer from left to right looking for a difference
  • Read decimal from left to right until there is a different digit

\[3010.111\color{yellow}{3}7 < 3010.111\color{yellow}{8}29\]

1.7 To round decimals

  • Round up by adjusting the digit to the left

1.8 To add and subtract decimals

  • Pad places to match digits
  • Subtract right to left

1.9 To multiply and divide decimals

  • handle the mantissa
  • adjust the exponents of 10

\[ 0.002 \times 0.004 = 2 \times 10^{-3} \times 4 \times 10^{-3} = 8 \times 10^{-6}\] \[0.000008\]

1.10 Solve the following challenge problems

  • What is the result in exponential form?

\[(0.001)^{-50}=?\]

  • What is the smallest whole number divisible by every whole number from 1 to 10?

  • The venom of a South American frog is so poisonous that 0.0000004 gm of the venom can kill a person. How many nanograms is the toxic dose?

  • The sum of each row, each column and each diagonal are all the same. Find the sum and the missing numbers.

\[\small\begin{matrix} \hline \left| \quad\ 1.75\ \right. &\left|\quad \ 3.12\ \right. & \left|\quad\quad\quad\ \right|\\ \hline \left|\quad \ 0.38\ \right. & \left|\quad\quad\quad\right. & \left|\quad\quad\quad\right| \\ \hline \left|\quad\quad\quad\right. &\left|\quad\quad\quad\right. & \left|\quad\quad\quad\right|\\ \hline \end{matrix}\]

2 Magic squares

In magic squares, the sum of each row, column, and diagonal are all the same. Calculate the sum and the values of a, b, c, d, and e.

\[\begin{matrix} \hline \left|\quad a \right. &\left|\quad \ \ 6 \right. & \left|\quad 11\ \right|\\ \hline \left|\quad b \right. & \left|\quad 10 \right. & \left|\quad \ c\ \ \right| \\ \hline \left|\quad d \right. &\left|\quad 14 \right. & \left|\quad \ e\ \ \right|\\ \hline \end{matrix}\]

3 Summary

\[x + 4\] same as

  • x plus 4
  • x increased by 4
  • the sum of x and 4
  • 4 more than x
  • 4 added to x

\[n - 3\] same as

  • n minus 3
  • n decreased by 3
  • the difference between n and 3
  • 3 less than n
  • 3 subtracted from n

\[\frac{3y}{4}\]

same as

  • (3 /4) times y
  • the product of (3/ 4) and y
  • (3/ 4) of y

\[\frac{5}{z}\]

same as

  • z divided by 5
  • the quotient of z and 5
  • z over 5

4 Solve these puzzles

4.1 Caffeine in beverages

  1. An 8-oz cup of black tea has 40 milligrams of caffeine. How does the concentration of green tea compare if green tea has 28 mg per cup? (in %)

  2. An 8-ounce cup of black tea has about 40 milligrams of caffeine, which is 10 times the amount of caffeine in a cup of decaffeinated tea. How much caffeine is in a cup of decaffeinated tea?

  3. What percent of the caffeine in tea is removed by the decaffeination process?

  4. An 8-ounce cup of brewed coffee has about 100 milligrams of caffeine. How much does this compare to the caffeine concentration in tea?

  1. Decaffeination removes about 97% or more of the caffeine in coffee beans. How many milligrams of caffeine in a cup of decaffeinated coffee?

  2. If Coke’s caffeine content is 34mg for a 12-oz can, how does the concentration of caffeine in coke compare to that of tea? (in %)

  3. If Diet Coke caffeine content is 46mg per 12 oz. how does this compare to the concentration of caffeine in regular coke? (in %)

  4. One ounce of dark chocolate contains about 24 mg of caffeine; milk chocolate contains one-quarter of that amount. If an 8 oz cup of Milo contains 1/2 ounce of milk chocolate, how many mg of caffeine does it contain?

5 ATT Stock Ticker tape

On 3 Jan 2023 an invester bought 100 shares of ATT stock at US$207.00 per share.

5.1 Daily ticker tape values

\[\small\matrix{\hline Days&1&2&3&6&7&8&9&10\\ \%chg&0.725&-0.959&-0.242&1.214&-1.199&2.184&0.475&-0.236\\ \hline Days&13&14&15&16&17&20&21&22\\ \%chg&0.948&0.000&-0.704&-2.600&2.184&3.088&1.152&-2.733\\ \hline Days&23&24&27&28&28&29&30&33\\ \%chg&2.108&1.835&-1.351&0.228&1.139&7.432&-0.629&-1.055\\ \hline Days&34&35&36&37&40&41&42\\ \%chg&-0.213&0.427&-0.213&-1.493&1.299&-1.282&2.381\\ \hline}\]

5.2 change plots

5.3 Daily Changes

5.4 Calculation

  • What is the current value of his investment?
  • If he sells his investment, what is his profit?
  • What is his average daily income over this period?

5.5 Investment Graph

6 Tiered pricing for electricity

  • 800 baht for 1st 1000 KWH (0-1000)
  • 1,200 baht / 1000KWH for next 1000 KWH (1001-2000)
  • 1,500 baht / 1000KWH for next 2000 KWH (2001-4000)
  • 2,000 baht / 1000KWH for over 4000 KWH

6.1 Calculating the cost

7 Volume discount

\[\small\matrix{Num&Desc&Q1& Q10 & Q100 & Q500\\ 1 & Microprocessor & 6.95 & 5.20 & 4.80 & 3.90 \\ 8 & RAM - 64 GB & 4.95 & 4.49 & 3.95 & 3.60\\ 4 & Motor Controller & 10.30 & 9.80 & 9.20 & 8.80 \\ 4 & Motor & 2.50 & 2.40 & 2.20 & 2.10\\ 1 & PowerSupply & 3.40 & 3.20 & 3.10 & 3.00 \\ \hline & Total unit & 101.15 & 93.12 & 85.1 & 78.4\\ }\]

PYU GE141 2022/1: GE 141 Mathematics for daily life [3]