Ratios Percents Proportions Scale Drawings & Scale Factors

RATIOS PERCENTS PROPORTIONS SCALE DRAWINGS & SCALE FACTORS

RATIO – A FRACTION THAT COMPARES LIKE THINGS CAN BE STATED AS

¾ 3:4 3 OUT OF 4 OR 3 TO 4

RATE – COMPARES UNLIKE THINGS USUALLY INVOLVES MONEY CAN BE STATED AS 6 COKES FOR $3.00 TO FIND THW COST OF EACH COKE DIVIDE THE TOTAL OF $ 3.00 BY 6 FOR A PRICE OF $.50

PROPORTION - A SET OF FRACTIONS THAT ARE MULTIPLES OF EACH OTHER FOR EXAMPLE ½ = 4/8 PROOF OF THE PROPORTION CAN BE GIVEN

FINDING A COMMON DENOMINATOR TO SHOW 2 EQUAL FRACTIONS
CROSS MULTIPLY THE FRACTIONS TO SHOW THE SAME PRODUCT ON BOTH SIDES OF THE = SIGN AS IN THIS CASE 1 X 8 = 2 X 4
CHANGING BOTH FRACTIONS TO A DECIMAL WHICH WOULD BE EQUAL OR A PERCENT SINCE BOTH WOULD BE .50 OR 50%
FIND A NUMBER THAT IS THE MULTIPLE OF BOTH FRACTIONS IN THIS CASE THE NUMBER WOULD BE 4

PERCENT- A PART OF A WHOLE THAT IS A FRACTION WHEN DIVIDED GIVES A DECIMAL WHICH IS MULTIPLIED BY 100 GIVES A PERCENT

¾ = .75 .75 X 100 = 75%

EVERY PERCENT THE FRACTION IS THE PERCENT GIVEN OVER 100 THEN REDUCED SO 80% = 80/100 = 4/5
IF A PERCENT HAS TO BE CHANGED TO A FRACTION AND HAS A FRACTION OR DECIMAL THAT MUST BE REMOVED BY MULTIPLYING THE FRACTION BY A NUMBR THAT RESULTS IN A WHOLE EX. 20.5% WOULD BE 2.5/100 IF THE NUMERATOR AND DENOMINATOR ARE MULTIPLIES BY 2 WILL FORM 5/200 WHICH CAN BE REDUCED TO 1/40
PERCENTS PROBLEMS CAN COME IN 3 FORMS 1. BASIC PERCENT OF A NUMBER 2. THE PERCENT IS GIVEN AND THE PART NEED THE WHOLE 3. THE PERCENT AND WHOLE ARE GIVEN NEED THE PART, ALL OF THESE CAN BE SOLVED BY A RATIO PROPORTION
THE KEY WORDS TO FORM THE PROPORTION ARE IS = PART AND OF = WHOLE
THE 10 % RULE TO TAKE 10 % OF ANY NUMBER MOVE THE DECIMAL BACKWARDS ONE PLACE SO 10 % OF 50 IS 5
TO CHANGE A DECIMAL TO % WHEN MULTIPLY BY 100 THE SAME AS MOVING THE DECIMAL FORWARD 2 PLACES

PERCENT OF A NUMBER

50% OF 40 = ______50/100 = X/40

100 X = 2000

DIVIDE BOTH SIDES BY 100 X = 20

FIND THE PERCENT WHEN GIVEN THE PART AND WHOLE

__% OF 20 = 10 X/100 = 10/20

20X = 1000

DIVIDE BOTH SIDES BY 20 X = 50

WHEN GIVEN THE PERCENT AND THE PART AND NEED TO FIND THE WHOLE

40 % OF _____ = 20 40/100 =20/X

40 X = 2000

THE PERCENT EQUATION

WHEN USE THE PROPORTION TO FIND THE PERCENT OF A NUMBER WILL DIVIDE BY 100 IN THE END WHICH IS THE SAME THING AS CHANGING THE PERCENT BACK TO A DECIMAL SO THE PERCENT OF A NUMBER CAN BE WRITTEN IN THE FORM OF AN EQUATION AS WELL AS A PROPORTION

P = .25 X 176 WHICH IS FOR 25 % OF 176 IS WHAT NUMBER

THE PROPORTION METHOD WOULD BE 25/100 = X/176

PERCENT ESTIMATION – ROUND THE PERCENT AND NUMBER TO THE NEAREST WHOLE OR 10 AND ESTIMATE NOT SOLVE USE THE 10% RULE TO ASSIST IN THIS

24% OF 200 24 CLOSE TO 25% SO 2 10S IN 20 10 % OF 200 = 20 SO 2 10S WOULD GIVE 40 AND 5 % IS HALF OF 10 SO 5% WOULD GIVE 10 SO 25% WOULD BE 50 OR 25% = ¼ SO IF 200 IS SPLIT INTO 4 EQUAL PARTS EACH PART WOULD BE 50

42 % OF 60 CLOSE TO 40% OF 60 10% OF 60 IS 6 4 10S IN 40 SO 4 X 6 = 24

MAKING PREDICTIONS – IF A SURVEY IS TAKEN THAT STATES A % OF THE POPULATION VOTED FOR SOMETHING THIS CAN BE APPLIED TO A LARGER GROUP AND THIS IS CALLED USING STATISTICS TO MAKE PREDICTIONS

IF 500 PEOPLE WERE POLLED AND 40% SAID THEY WANTED A STARBUCKS IN THE AREA AND THE TOWN HAD A POPULATION OF 6000 PEOPLE HOW AMNY PEOPLE WOULD YOU PREDICT WOULD WANT A STARBUCKS TO COME?

40/100 = X/ 6000 WHICH WOULD GIVE ( 40 X 6000) = ( 100 TIMES X)
SO 240000 = 100X SO X = 2400 PEOPLE WOULD WANT STARBUCKS OUT OF 6000 PEOPLE

PERCENT OF CHANGE – THIS MEANS SOMETHING HAS CHANGED SO IT IS DIFFERENT THE CHANGE CAN BE AN INCREASE OR A DECREASE IN BOTH CASES ( THE DIFFERENCE BETWEEN THE 2 NUMBERS) / THE ORIGINAL NUMBER WHICH WILL GIVE A DECIMAL THAT DECIMAL X 100 GIVES THE PERCENT OF CHANGE

IF A STUDENT SCORED A 70 ON THE FIRST TEST AND A 90 ON THE SECOND TEST WHAT WAS THE PERCENT OF INCREASE IN THE STUDENT SCORE
( 90 – 70 )/70 X 100 = 20/70 = .28 X 100 = A 28 % INCREASE IN THE STUDENT SCORE

IF A COAT SOLD FOR $150.00 AND IS NOE ON SALE FOR $95.00 WHAT IS THE SALE DISCOUNT OF THE COAT?
( 150 – 95 ) / 150 = 55/150 = .36 X 100 = 36 % OFF IS THE DISCOUNT

SALES TAX AND DISCOUNT:

IN ORDER TO SOLVE THESE PROBLEMS YOU MUST FIRST DETERMINE THE PERCENT OF THE NUMBER USING THE RATIO METHOD ABOVE

SALES TAX

ONCE YOU FIND THE PERCENT PART ADD THAT AMOUNT TO THE PRICE TO OBTAIN THE FINAL PRICE

EX. IF YOU BUY A $20.00 GIFT CARD AND THE SALES TAX RATE IS 5% WHAT IS THE FINAL PRICE?

5/100 = X/20 ( 5 X 20) = 100X X = $1.00 SO THE FINAL PRICE IS $21.00

DISCOUNTS-

ONCE YOU FIND THE PERCENT OF THE NUMBER SUBTRACT THAT AMOUNT TO FIND THE FINAL PRICE

EX.: IF A TV SELLS FOR $500.00 AND YOU HAVE A SALE OF 20% OFF WHAT IS THE COST OF THE TV BEFORE TAXES?
20/100 = X/500 ( 20 X 500 ) = ( 100 x X ) AFTER DIVIDE X = $100.00 SO THE FINAL PRICE IS 500.00 -100.00 = $400.00

SIMPLE INTEREST

THIS IS USED TO DETERMINE THE AMOUNT OF INTEREST MONEY WILL EARN WHEN INVESTED IN A BANK
I = PRT I = INTEREST P = PRINCIPLE R = RATE WHICH THE % MUST BE CHANGED BACK TO ITS DECIMAL FORM T = TIME AND MUST BE IN YEARS

EXAMPLE: IF $1000.00 IS INVESTED FOR 5 YEARS AT 10% WHAT WILL THE INTEREST EARNED AT THE END OF 5 YEARS AND THE TOTAL AMOUNT OF MONEY IN THE BANK?

I = 1000 X .05 X 5 I = $500.00 SO THE TOTAL WOULD BE $1500.00

THE FORMULA CAN BE USED TO SOLVE FOR ANY PART

AS IN THE CASE THE INTEREST EARNED WAS $500.00 AND $1000.00 WAS INVESTED FOR 5 YEARS WHAT WAS THE INTEREST RATE?

I = PRT 500 = 1000 x R x 5 500 = 5000R TO SOVLE FOR R = .01 x 100 10%
OR CAN USE THE FORMULA WITH GUESS AND CHECK

SCALE DRAWINGS

A MODEL DRAWING THAT USES A SCALE FACTOR TO DRAW AN OBJECT LARGER OR SMALLER THAN THE ACTUAL AS IN THE CASE OF HOUSE PLANS AND MAPS

SCALE

THE RATIO THAT COMPARES THE MODEL TO THE ACTUAL MAY USE FRACTIONS DECIMALS AND DIFFERENT UNITS

SCALE FACTOR

THE RATIO THAT COMPARES THE MODEL TO THE ACTUAL WITHOUT ANY FRACTIONS OR DECIMALS IN THE MOST REDUCED FORM AND ALL UNITS MUST BE THE SAME
IF YOU HAVE A TRIANGLE AND THE 2 LONG SIDES ARE 8 INCHES ON A MODEL AND THE BASE IS 4 INCHES

AN IF THE SCALE IS 2 INCHES = 2 FEET

WHAT AND THE ACTUAL

MEASUREMENTS OF THE TRIANGLE AND THE

SCALE FACTOR OF THE SCALE?

2 INCHES/ 2 FEET = 8 INCHES/ X FEET

( 2 x X ) = 8 x 2 ) = 2X = 16 X = 8 FEET

2 INCHES/ 2 FEET = 4 INCHES/ X FEET

( 2 x X ) = ( 4 x 2 ) = 2X = 8 X = 4 FEET

SCALE FACTOR 2 INCHES = 2 FEET
REDUCE TO I INCH – 1 FOOT THEN CHANGE TO THE SAME UNITS
SO I INCH= 12 INCHES IS THE SCALE FACTOR

DILATION-

A DRAWING IS PLOTTED ON A 4 QUADRANT GRID AND A FACTOR IS MULTIPLIED TIMES EACH X AND Y TO ENLARGE OR SHRINK THE OBJECT
IF THE FACTOR IS LESS THAN 1 THE OBJECT SHRINKS IF MORE THAN ONE THE OBJECT ENLARGES

IF THE TRIANGLE ABOVE HAD THE ORDERED PAIRS 1,0 6,6 12, 0 AND THE SCALE FACTOR OF 2 WAS APPLIED TO THE TRIANGLE
THE NEW ORDERED PAIRS WOULD BE 2,0 12,12 24,0 AND THE TRIANGLE WOULD ENLARGE
IF THE SCALE WAS ½ THE NEW PAIRS WOULD BE .5,0 3,3 6,0