IT100 Session 3: - Mathematical operations

Unit 1. Order of Precedence

So what is the answer?

\[\large\begin{array}{rclr} 3 \times 4 - 2 \times 5 & = & (3\times 4)-(2\times 5) &(1)\\ (3 \times 4) - (2 \times 5) & = & 12 - 10 = 2 &(2)\\ \left((3 \times 4) - 2\right) \times 5 & = & (12 -2)\times 5 = 50 & (3)\\ 3 \times (4 - 2) \times 5 & = & 3\times 2\times 5 = 30 & (4)\\ 3 \times \left(4 - (2 \times 5)\right) & = & 3\times (4 - 10)= -18 & (5)\\ \end{array}\]

Challenge questions

\[\large\begin{array}{rclr} 10 - 2^2 + 3 &=& 10 - 4 + 3= 9 &(1)\\ & & \\ {3^2 \times 7 - 3 + 6 \over 4 + 2} &=& {(9\times 7)- 3 + 6\over 6} = 66 \div 6 = 11 & (2)\\ \end{array}\]

Unit 2: Exponents

Challenge questions

Simplify the following:

\[\begin{array}{rclr} {1 \over \sqrt{x}} & = & {1\over x^{1\over 2}} = x^{-{1\over 2}} & (1)\\ {2^x \times 8^2 \over 16} & = & {2^x \times (2^3)^2 \over 2^4} = 2^x \times 2^6 \times 2^{-4} = 2^{(x+2)} & (2)\\ \end{array}\]

3. The human heart beats about 70 times per min. Express in scientific notation, the number of times a heart would be expected to beat in 72 years.

\[\begin{array}{l} {70\over min} \times {60 min\over hr} \times {24 hr\over day} \times {365 days\over yr} \times {72 yr} &\\ \qquad = 7\times 10^1 \times 6 \times 10^1 \times 2.4 \times 10^1 \times 3.65 \times 10^2 \times 7.2 \times 10^1 \\ \qquad = 7\times 6 \times 2.4 \times 3.65 \times 7.2 \times 10^{(1+1+1+2+1)} \\ \qquad = 26490 \times 10^{6} = 2.649 \times 10^4 \times 10^6 \\ \qquad = 2.649 \times 10^{10}\\ \end{array}\]

Unit 3: Fractions

\[\large\begin{array}{rclr} 2{2\over 3} \times {1\over 5} + 1{1\over 2} \div 2{1\over 2} & = & {8\over 3 \times 5} + {{3\over 2}\over {5\over 2}} = {8\over 15} + {9\over15} = {17 \over 15} & (1)\\ & & \\ {{\left({1\over 3} \right)^2}\times 4{1\over 2}\over 3} & = & {{1\over 9} \times {9\over 2}\over 3} = {2 \times {1\over 2}\over 2 \times 3} = {1\over 6} & (2)\\ \end{array}\]

Unit 4: Identity

## Problem of Identity

Give an example and determine the value of \(\large I\) for each relationship.

\[\large\begin{array}{rlrcrlr} a \times I = a & I = 1 & (1) & & {a \over I} = a & I = 1 & (2)\\ a - I = a & I = 0 &(3) & & a + I = a & I = 0 &(4)\\ a^I = a & I = 1 &(5) & & a \times a^I = a & I=0 & (6)\\ \end{array}\]

Unit 5: Reciprocal

Determining the reciprocal

Identify the reciprocal for each relationship.

\[\large\begin{array}{rlcrl} a \times I = 1 & I = {1\over a} & (1) & & {a \over I} = 1 & I = a & (2)\\ a - I = 0 & I = a & (3) & & a + I = 0 & I = -a & (4)\\ a \times a^I = 1 & I = -1 & (5) & & a \times a^I = a & I = 1 & (6)\\ \end{array}\]

Unit 6: Basic laws of Math

Producing a trail mix

Recipe

Volume	Ingredient	Cost / 250 gm
100 gm	roasted almonds	$4.00
100 gm	roasted pecans	$6.50
100 gm	roasted cashews	$4.75
100 gm	sunflower seeds	$0.40
100 gm	pumpkins seeds	$0.75
150 gm	dried cherries	$3.90
200 gm	raisin	$5.60
100 gm	cranberries	$6.90
200 gm	chocolate chips	$3.50
10 gm	sea salt	$0.35
5 gm	cinnamon	$8.00
5 gm	grown nutmeg	$9.50
1170 gm	Totals	-

Challenge Questions

• What is the cost per kilo?

\[{19.30\over 1170} = 16.50 \qquad \qquad(1)\]

• If the market value is US$25.00 per 500 gm, what is the profit margin?

\[\hbox{Income:}\quad {25\over 500gm} = {50\over 1kg} \qquad\qquad (2)\] \[\hbox{Cost:}\quad 16.50\qquad\qquad\qquad\] \[\hbox{Profit:}\qquad 50.00 - 16.50 = 33.50 \] \[\hbox{Profit margin:}\quad {33.50 \over 50} = 67\%\]

Pound Cake

\[\large\begin{array}{lrr} Ingredient & Amt/ kg & Cost/kg\\ Butter & 1 & 2.5 \\ Eggs & 1 & 3.6 \\ Flour & 1 & 1.5 \\ Milk & 1 & 3.8 \\ Sugar & 1 & 1.8 \\ \hfill Totals & 5 kg & 13.2 \\ \hfill Cost/kg & & {13.20 \over 5} = 2.64 \\ \end{array}\]