2 are both ways of saying a ratio of 2 to 5

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1 Unit 4 Ratios A Ratio is a comparison of two related quantities. Ratios are expressed in two forms. 2 : 5 or 5 2 are both ways of saying a ratio of 2 to 5 1. Conversion factors are ratios. Express 100 cm = 1000mm as a ratio Ratios should be reduced to lowest terms (same as reducing fractions) 2. Express 5280 ft = 1610 m as a ratio Ratios that include units are correct in either order 3. Gear Ratios relate the number of teeth on one gear, to the number of teeth on the meshed gear. The gear ratio for a set of gears is 5:1. If the first gear has 100 teeth how many teeth are on the second gear? (gear to pinion) 4. Mixing ratios give a ratio for combining components or ingredient to get a desired product. This is basically a recipe. Rice Krispy Treats ¼ cup margarine (4 TBSP) 40 regular Marshmallows 5 cups of cereal What is the ratio of marshmallows to cereal? Of Marshmallows to margarine? 5. To keep the men in the class happy consider the mixing ratio for concrete. The mixture varies according to the use and strength desired in the product. The in 1984 DOT used this guide for Class C concrete. (Class C concrete has a compressive strength of 2300 psi) pounds per cubic yard of Class C Concrete Portland Cement dry fine aggregate dry No 2 course aggregate Write a ratio to describe this mixture Find the amount of sand, gravel and Portland cement needed to mix 5 cubic yard of Class C concrete. 1

2 Direct Proportions Proportions are equal fractions Reducing fractions produces a proportion This could also be written as equal ratios To use proportions are used to find values. Begin with the fraction in words Set up an equation with two equal fractions. 6. The ratio of primary turns to secondary turns in a transformer is equal to the ratio of primary voltage to the secondary voltage. If the primary coil at 120v has 5000 turns and the secondary coil has 500 turns what is the voltage of the secondary coil? Draw a picture Set up the proportion as a fraction in words then fill in the numbers. Solve by "cross multiplying" 7. Car suspension for a particular car is a directly proportional relationship. The new toe-in setting for a different diameter wheel with the same toe angle can be calculated using a direct proportion. Set up the proportion of Toe in setting to diameter. Find the new toe-in setting if a 16" diameter wheel is used to replace a 24" diameter wheel with a toein setting of 3/32" Set up the proportion as a fraction in words then fill in the numbers. Solve by "cross multiplying" 2

3 Similar Polygons Similar polygons have the same shape but not same size. This means that these figures have the same number of sides and the angles and the sides are proportional 8. Find the lengths of the missing side 15.0cm 25.0cm 18.0cm x Inverse Proportions Playing marbles is good way to observe an inverse variation relationship at work. A large marble moving slowly has the same momentum as a small marble moving quickly. In an inverse proportion relationship, one value increases as the other decreases. If a large marble hits a small marble then you would expect If a small marble hits a large marble you would expect Why is it cheating to use a large ball bearing when playing marbles? The easiest way to set up an inverse proportion equation is as a constant product. Consider the two ways to set up an equation involving momentum As a constant product m 1 v 1 = m 2 v 2 or as an inverse proportion m 1 v = 2. m 2 v 1 9. A 200g stationery marble is hit by a 500g marble traveling at 15 cm/s. After impact the large marble is traveling at 3 cm/s and the smaller marble is also moving. Assuming a perfectly elastic impact (all the momentum is conserved), what is the velocity of the smaller marble after impact? 3

4 Torque is calculated as T = F x d. The x as a multiplication symbol indicates that this is a cross product where the force is applied perpendicular to the distance from the point of rotation. For a constant torque, force and distance are inversely proportional or a one increases the other decreases. A Lever and Fulcrum is really just a Torque problem. The Torque produced on the short side of the lever is equal to the Torque applied to the long side of the lever. 10. Calculate the force required to lift a 100 lb rock if the fulcrum is placed 2 feet away from the rock and a 6 ft board is used as a lever. 11. Take look at the Tow truck pictures on the web. What are the two ways that the mistakes of the tow truck drivers could have been avoided in pulling the car out of the water? There is an inverse variation relationship between two rotating pulleys connected by a belt. Velocity of the belt is constant, and the larger the pulley the slower it must rotate. The diameter and rpm are inversely proportional. For a simple pulley and belt system this reduces to (diameter)(rpm) = (diameter)(rpm) D 1 ω 1 = D 2 ω If a 15 in diameter pulley is connected to and 8 in diameter pulley turning at 50 rpm, what is the rpm of the larger pulley? A similar relationship can be written for gears but use teeth and rpm instead. Write the relationship Why use teeth instead of diameter? 13. A large gear with 50 teeth turning at 25 rpm is meshed with a pinion with 10 teeth. What is the rpm of the pinion? 4

5 Consider the General Law for Perfect Gases Combined Variation: PV T =. 1 P V T This relationship involves both a direct and inverse variation relationship. 2 Note that it is important to use absolute values for both P and T when applying this law Temperature in Kelvin or Rankine P in psia or kpa Kelvin = Celsius Rankine = Fahrenheit absolute pressure = gauge pressure + 1 atmosphere 1 atmosphere = 14.7 psi or 101.3kPa Use the Gas law in the following problems 14. If m 3 of air enters an air compressor that begins its cycle at normal atmospheric pressure (an absolute pressure of kpa) and leaves the compressor at the same temperature (isothermal compression) but with an absolute pressure of 3250 kpa, what is the volume of the air after compression? 15. The absolute pressure in an oxygen cylinder is 1724 kpa when the temperature is 30 C, what will the absolute pressure be if the cylinder is cooled to 10 C (Change to temperature to Kelvin) 5

6 Percent means parts per 100 so 1% is 1 part of a 100 or Unit 5 Percent 1 12 = and 12% is 12 parts of 100 or = Percents are commonly used in interest rates, tolerances, safety factors, estimates etc. It is important to deal with these successfully. Percents are express in percent notation, as fractions and as decimals. You need to be able to change forms at will To change from a fractional part to percent notation, find the decimal equivalent and move the decimal 2 places to the right To change from percent notation to a decimal move the decimal point two places to the left 1) 13% 78% 9% 3 8 To change percent notation to a fraction, place the percent over 100 2) 13% 78% 9% Percent applications use the relationship Percent Applications Percent = Part Original or in another form Part = percent of the Original or Part = %(Original) The most difficult part of percent problems is to correctly identify the part and the Original. The original is the starting value, the part is portion you are comparing with the original (Note the author uses different words here. This may be confusing) 16. You get a 60 raise to $15.60 an hour. What percent raise is this? 17. A heating and cooling contractor places a bid on a job based on parts plus 75% for labor and overhead plus 8% sales tax. Find the bid on a job where the cost of parts is expected to be $

7 Percent Change The percent change involves an increase or decrease in the original value. The change is compared to the original value There are two ways to approach these problems. A. Find the change of the original value =%change x original: then add or subtract to get the final value. Or B. Add or subtract the percent change from 100% to find the final percent and then calculate the part left using the formula Part = percent original 18. You are scheduled to receive a 5% raise on your current pay rate of $15.00 an hour. What is your new pay rate? 19. A certain 20 amp fuse will take a current of 22 amps for a short period. What percent overload is this? Efficiency Efficiency of a motor is the output power as a percent of the input power or output Efficiency =. input Remember that to calculate percents the units in the fraction must all be the same. All motors have an efficiency of less than 100%. Work is lost to heat and friction. Calculate the efficiency of these motors 20. A 1 hp belt sander draws 8.5 amps at 120v. What is the efficiency of this motor Some important relationships Electrical Power P = IV in watts 1 Hp = watts 21. A motor draws 17.3 amps at 120v to produce 2 ¼ horsepower. What is the efficiency of this motor? 7

8 22. A waterfall is144 ft high has a flow rate of 50cfm. If 50% of the power can be utilized in powering a hydroelectric generator, what is the generator's output in horsepower? 1 Hp = ft-lb/min Water weighs 62.4lb/ft 3 Coefficient of performance Coefficien t of performance = β = heat work rejected required Heat pumps and refrigerators can have a coefficient of performance or ß that exceeds 100%. Why? 23. Find the coefficient of performance of a 10 ton refrigeration system that requires a 15 hp compressor. 1 ton of refrigeration capacity = 200 Btu/min 1 hp = 33,000 ft-lb/min 1Btu= 778 ft-lb 8

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