MGF 1107 Exam 2
Example: You arrive in Italy with $700 that you want to convert into euros. If the conversion rate is $1 = .9 €, how many euros will you have?
$700 × .9€/$1=630€
5 years after it's bought, a certain baseball card is worth 32% more than its original value. Its new value is 132% of its original value.
(100 + 32)% = 132%
If Josh earns 50% per hour more than Janet does, then Josh's salary is 150% of Janet's.
(100 + 50) = 150%
If Town A's population is 20% less thanTown B's, then Town A's population is 80% of Town B's.
(100 - 20)% = 80%
A shirt is on sale for 30% off. The new price is 30% less than the original price, so the new price is 70% of the original price.
(100 - 30)% = 70%
Example: Find the rate of inflation from 1993 to 1994.
(CPI_1994 - CPI_1993)/CPI_1993 × 100% = (148.2 -144.5)/144.5 × 100% = 2.56%
RECAP
1) Index number is a comparison of values. index number = value/(reference number) × 100 It is expressed as a number (not a percent). 2) You can find a different year's value by using the given value and the index number. new value = index number x reference value (as a percent) 3) Rate of inflation is just the average change of the CPI over time. (CPI_new - CPI_old)/CPI_old × 100% It is expressed as a percent. 4) You can translate a value from one year into what it would be in dollars for a different year. new value in $Y = (old value in $) x CPI_new/CPI_old
You purchase a car for $24,000 and two years later it is worth $18,000. What is the relative change of the value of the car over the 2 years?
1. Find the absolute change: 18,000 - 24,000 = -6,000 2. Find the relative change: (-6000)/24000 × 100%=-25% The value of the car has decreased by 25%.
Example: You buy a house in 2005 for $165,000. When you go to sell it in 2010, you are able to get $235,000. What is the relative change of the house value?
1. Find the absolute change: 235,000 - 165,000 = 70,000 2. Find the relative change: 70000/165000 × 100%=42.42% The value of your house has increased by 42.42%.
Example: If a movie ticket cost $9 in 1999, what was its price in 2012?
1. Find the index number. Use CPI1999 = 166.6 and CPI2012 = 229.6 CPI_2012/CPI_1999 = 229.6/166.6=1.38 2. Find the new price. 1.38 x $9 = $12.42 in 2012
If a quantity doubles in value, what is its relative change?
1.Find the absolute change: 2x -x=x 2. Find the relative change: x/x × 100%=100% When a quantity doubles in value, its relative change is 100%
Example: A man in Florida earns $40,971 a year while a woman earns $34,768. What is the relative difference of women's pay in comparison to men's?
1.Find the absolute difference: 34,768 - 40,971 = -6,203 2. Find the relative difference: (-6203)/40971 × 100%=-15.14% Women in Florida earn about 15% less than their male counterparts.
SAY WE SWITCH THE PROBLEM AROUND... Example: A man in Florida earns $40,971 a year while a woman earns $34,768. What is the relative difference of men's pay in comparison to women's?
1.Find the absolute difference: 40,971 - 34,768 = 6,203 2. Find the relative difference: 6203/34678 × 100%=17.814% Men in Florida earn about 18% more than their female counterparts.
Example: A typical dose of Benadryl is 25 mg every 6 hours for a 100-pound person. How many 12.5mg chewable tablets would be taken in a week?
1tablet/12.5mg × 25mg/6hr × 24hr/1day × 7day/1week=56 tablet/week
Example: Convert 25km/min to m/hr.
25km/1min×0.62137m/1km×60min/1hr= 932.06 m/hr
Example: Your utility company charges 13¢ per kilowatt-hour of electricity. What is the daily cost of keeping lit a 75-watt bulb for 12 hours each day?
75watt × 1kilowatt/100watt × 12hr= .9 kilowatt-hr → .9 kilowatt-hr × 13¢/(1kilowatt-hour)=11.7¢
PERCENT
A fraction of 100
Example: In 1980, it cost 25 cents to buy a can of Coke. If the price index for 1999 is 420, how much did a can of Coke cost in 1999?
A price index of 420 means the price of a can in 1999 was 420% of the price in 1980. 420% x $.25 = 4.2 x $.25 = $1.05
INDEX NUMBER
A simple way to compare measurements at different times or in different places. index number = value/(reference number) × 100 The reference value is the chosen value from a particular time or place. THE MAIN PURPOSE OF AN INDEX NUMBER IS TO MAKE COMPARISONS.
UNIT CONVERSION
Allows us to switch from one unit to another. How many inches are in a yard? 1yd × 3ft/1yd × 12in/1ft
IMPORTANT TERMS
Concentration - The amount of one substance mixed with another. Air Pollutant - number of molecules or pollutant per million molecules of air Medicine Dosages - recommended concentration per kilogram of body weight Blood Alcohol Content - units of grams of alcohol per 100 milliliters of blood
IMPORTANT TERMS
Density - Compactness or crowding. Material Density - mass/weight per volume Density of water is 1 g/cm^3 Population Density - number of people per unit area ppl/mi^2 Information Density - memory per unit area gb/in^2
UNIT
Description of what a quantity measures/counts
IMPORTANT TERMS
Energy - What makes matter move or heat up. Joule - The international metric unit of energy. Power - The rate at which energy is used. energy/time Watt - The international metric unit of power. 1 watt = 1 joule/sec Kilowatt-Hour - A specific unit of energy. 1 kilowatt hour = 3.6 million joules
We can use the rate of inflation to see how much money from a certain year would translate to money in a different year.
Example: A US senator made $44,600 in 1976. How would that translate into 2015 dollars? To adjust prices for inflation from year X to year Y price in $Y = (price in $X) x CPI_Y/CPI_X price in $Y = (price in $X) x CPI_Y/CPI_X price in 2015 = $44,600 x 237.017/56.9 price in 2015 = $185,781.34 In reality, a senator made $174,000 in 2015.
CONVERTING TO PERCENTAGES
From a Fraction: Divide the denominator into the numerator and then move the decimal 2 spaces to the right and add the %. ¾ → .75 → 75% From a Decimal: Move the decimal two spaces the right and add the %. 1.35 → 135%
OF VERSUS MORE THAN/LESS THAN
If the new value is P% more than the reference value, it is (100 + P)% of the reference value. 21% more than → 121% of 92% more than → 192% of If the new value is P% less than the reference value, it is (100 - P)% of the reference value. 12% less than → 88% of 42% less than → 58% of
THE MAIN IDEA:
Increase: Target = (100 + p)% x old Decrease: Target = (100 - p)% x old
OF VERSUS MORE THAN/LESS THAN
More/Less Than: States the relative change. A population triples in size from 1000 to 3000. relative change = (3000 -1000)/1000 × 100%=200% The new population is 200% more thanthe original population.
What conversions would you need to go from km/min to m/hr?
Need to convert kilometers to miles: km/m or m/km? m/km because km is in the numerator. Need to convert minutes to hours: min/hr or hr/min? min/hr because min is in the denominator. km/min×m/km×min/hr
KEY WORDS IN UNIT ANALYSIS
OF - Multiplication '60% of 20' means 60% × 20 PER - Division '20 miles per hour' means 20 miles ÷ 1 hour SQUARE - To the Second Power '12ft × 10ft = 120 ft2 or 120 square feet' CUBE - To the Third Power '3ft × 2ft × 4ft = 24 ft3 or 24 cubic feet'
OF VERSUS MORE THAN/LESS THAN
Of: Ratio of the new value to the old value. A population triples in size from 1000 to 3000. (target population)/(old population)= 3000/1000=3=300% Its new population is 300% of the original one.
RECAP
Percentage Points: absolute change or absolute difference target/compared value - reference value Percentage: relative change or relative difference absolute change/difference x 100% reference value
UNIT ANALYSIS
Process of working with units to help solve problems.
USING PERCENTAGES TO DESCRIBE CHANGE
Reference Value: The original value we start with. Target Value: What your reference value becomes. Absolute Change: The actual increase or decrease. absolute change = target value - reference value Relative Change: Ratio/comparison of the absolute change to the reference value. Often expressed as a percentage. relative change = (absolute change)/(reference value) × 100%
USING PERCENTAGES FOR COMPARISONS
Reference Value: The value being used as a basis for comparison. (Comes after the word than) Compared Value: The new value you're looking at. Absolute Difference: The difference between the reference value and the compared value. absolute difference = compared value - reference value Relative Difference: Ratio/comparison of the absolute difference to the reference value. Often expressed as a percentage. relative difference = (absolute difference)/(reference value) × 100%
RELATIVE VS ABSOLUTE CHANGE
Relative Change is always the difference over the original value. Absolute Change: expressed in percentage points Relative Change: expressed withpercent sign
USING PERCENTAGES FOR COMPARISONS
Relative difference is always the difference over what you are comparing to. Absolute Difference: expressed in percentage points Relative Difference: expressed with percent sign
PROBLEM SOLVING WITH UNITS
STEP 1: Identify the given units and the desired units. STEP 2: Use both set of units to find a strategy for problem solving. You can ONLY multiply or divide different units. You cannot add or subtract them. It is easier to multiply by the reciprocal than divide. STEP 3: Check your answer to make sure your units are correct.
FOR 3E
STUDY FROM PPT.
CONSUMER PRICE INDEX (CPI)
The CPI is an average of prices for a sample of more than 60,000 goods, services and housing costs. The CPI is computed and reported monthly. The CPI allows us to compare prices at different times. The CPI helps us to measure inflation.
RATE OF INFLATION
The rate of inflation from one year to the next is the relative change in the CPI. (CPI_new - CPI_old)/CPI_old × 100%
CONVERTING FROM PERCENTAGES
To a Fraction: Put over 100 and simplify. 25% → 25/100 → ¼ To a Decimal: Drop the % and move the decimal 2 spaces to the left. 3.2% → .032
Example: A cube of wood measures 3 cm on a side and weighs 20 grams. What is its density?
Volume is 27cm3 → Density is 20gm/(27cm^3 ) = .74 gm/cm^3 Will it float in water? Yes, since it less dense than water.
USING PERCENTAGES TO DESCRIBE CHANGE
We can use percentages to describe how quantity changes over time. This change may be an increase or decrease. Ex. Population typically increases over time.
Example: The cost of an iMac today is $1799. Using the 2000 price of $799 as the reference value, find the price index for an iMac today.
index number = (current price)/(2000 price) × 100 = 1799/799 × 100 = 225.26 The current iMac cost is 225.26% of the 2000 cost.
