PERT

Fractions

A fraction consists of a numerator (top) and a denominator (bottom).

Form	Description	Examples
Properfraction	Numeratorsmallerthanthedenominator	3/5 7/12 ,
Improperfraction	Numeratorequaltoorlargerthanthedenominator	17/5 9/9 ,
Mixednumber	Awholenumbercombinedwithaproperfraction	32/5

Improper → mixed: 17/5

17 ÷ 5 = 3 remainder 2 → 3 2/5

Mixed → improper: 4 3/7

(4 × 7) + 3 = 31 → 31/7

The quotient becomes the whole number and the remainder becomes the new numerator; the denominator does not change.

Equivalent and Simplified Fractions

Equivalent fractions represent the same value. Multiply or divide the numerator and denominator by the same nonzero number: 2/3 = 4/6 = 6/9 = 8/12. To simplify, divide both by their GCF: for 24/36 the GCF is 12, so 24/36 = 2/3.

Operations with Fractions

Operation	Procedure	Example
Add/subtract	Findtheleastcommondenominator(LCD) convert ,	2/3+1/4=8/12+3/12=11/12
	eachfraction thenaddorsubtractnumerators ,
Mixednumbers	Converttoimproperfractions orworkwithwhole ,	51/4−23/4=45/4−23/4=22/4=21/2
	andfractionpartsseparately;borrowingmaybe
	neededwhensubtracting
Multiply	Multiplynumeratorstogetheranddenominators	3/4×2/5=6/20=3/10
	together;simplifybeforemultiplyingwhenpossible
Divide Multiplybythereciprocalofthedivisor 3/4÷2/5=3/4×5/2=15/8=17/8 1 ecimals ecimalsarefractionswhosedenominatorsarepowersof10:045=45/100=9/20 Toconvertafractiontoadecimal . . , ividethenumeratorbythedenominator:3/8=3÷8=0375 . .

Decimals

Decimals are fractions whose denominators are powers of 10: 0.45 = 45/100 = 9/20. To convert a fraction to a decimal, divide the numerator by the denominator: 3/8 = 3 ÷ 8 = 0.375.

Operation	Procedure	Example
Add/subtract	Alignthedecimalpoints	47+035=505 . . .
Multiply	Multiplyaswholenumbers;theproducthasasmany	06×007=0042(1+2=3places) . . .
	decimalplacesasthefactorscombined
Divide	Movethedivisor'sdecimalpointtomakeitawholenumber;	144÷012=144÷12=12 . .
	movethedividend'sdecimalpointthesamenumberof
	places

Comparing Rational Numbers

Convert all numbers to the same form (decimals, or fractions with a common denominator) before comparing. For two fractions, cross-multiply: to compare a/b and c/d, compare ad and bc.

Compare 5/8 and 7/11

5 × 11 = 55 8 × 7 = 56

55 < 56, so 5/8 < 7/11

Order 0.6, 5/8, 0.58 from least to greatest

5/8 = 0.625 → 0.58 < 0.6 < 5/8

Write each cross product on the side of the fraction whose numerator it uses.

Ratios and Rates

A ratio compares two quantities. The ratio of 3 to 5 can be written as 3:5, 3 to 5, or 3/5, and ratios simplify like fractions: 12:18 = 2:3. A rate is a ratio of quantities with different units, and a unit rate has a denominator of 1. Traveling 240 miles in 4 hours is a rate of 240 miles/4 hours, a unit rate of 60 miles per hour.

For a part-to-part ratio, add the parts to get the whole. If boys to girls is 2:3 in a class of 35, there are 5 parts of 35 ÷ 5 = 7 students: 14 boys and 21 girls.

Proportions

A proportion states that two ratios are equal. In a/b = c/d the cross products are equal: ad = bc. This property solves for an unknown value.

3/5 = x/20

3 × 20 = 5 × x

60 = 5x

x = 12

Set up the proportion with matching units in the same positions (for example, miles over hours on both sides).

Direct and Inverse Variation

The quantities increase or decrease together at a constant

One quantity increases as the other decreases; the product

rate; k is the constant of proportionality and the graph is a line

stays constant. If 4 workers complete a job in 6 hours, the job

through the origin. If y = 15 when x = 3, then k = 15/3 = 5 and y

takes 4 × 6 = 24 worker-hours. With 3 workers, the time is

= 5x.

24/3 = 8 hours.

Percent

Percent means per hundred. To convert a percent to a decimal, divide by 100 (move the decimal point two places left): 75% = 0.75. To convert a decimal to a percent, multiply by 100: 0.32 = 32%. To find a percent of a number, multiply the decimal form by the number: 25% of 80 = 0.25 × 80 = 20.

part = percent × whole

Every basic percent problem is this equation with one value missing; rearrange as needed.

Find the…	Example	Solution
Part	Whatis30%of50?	030×50=15 .
Percent	15iswhatpercentof60?	15÷60=025=25% .
Whole	20is40%ofwhatnumber?	20÷040=50 .

Percent Increase and Decrease

percent change = (new − original) ÷ original × 100

A price that rises from 40 to 52 changes by 12, so the increase is 12/40 × 100 = 30%. Always divide by the original amount.

For successive percent changes, calculate each step on the result of the previous step; percents are never simply added. A multiplier makes each step one operation: an increase of r% multiplies by (1 + r/100), and a decrease multiplies by (1 − r/100).

Start at 100; increase 20%, then decrease 10%

Step 1: 100 × 1.20 = 120

Step 2: 120 × 0.90 = 108

Net result: 108, an 8% increase (not 10%)

Equivalent single multiplier: 1.20 × 0.90 = 1.08.