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-2 ^ 2
- Even exponent: always positive result
-2 ^ 3
- Odd Exponent: Alwys negative result
A businesswoman spends 1/5 of her income for rent, and 3/8 of
the remainder goes towards salary. How much does she have remaining.
- 1. 1/5 of her income gone leaves 4/5
- 2. 4/5 * 3/8: 12/40
- 3. Simplify
a purchaser paid $17.16 for an article that had recently been increased in price by 4% what was the price of the article before the increase?
- 1. Increased in price by 4% means 17.16 is 104%
- 2. Price before then was 96% of this price
- 3. .96 * 17.16: 16.50
A naval detachment has enough rations to feed 16 people for
10 days. If 4 more people join the detachment, for how many fewer
days will the rations last?
- 1. 16 people for 10 days means 160 total rations
- 2. 4 more people means 20 people now
- 3. divide 160 rations amongst 20 people
- 4. lasts 8 days
- 5. 8 is 2 fewer than 10
A field can be plowed by 9 machines in 5 hours. If 3 machines
are broken and cannot be used, how many hours will it take to plow
- 1. 9 machines in 5 hours means 45 total machine hours to get the job done
- 2. 3 broken means 6 functioning
- 3. divide 45 hours of work among 6
- 4. It'll take 6 machines 7.5 hours
In a 3 hour examination of 320 questions, there are 40
mathematics problems. If twice as much time should be allowed for
each mathematics problem as each of the other questions, how many
minutes should be spent on he mathematics problems?
- 1. 40 math questions, 320 total means 280 non-math
- 2. 40x + (1/2)280x = 180 minutes
- 3. 40x + 140x = 180x = 180
- 4. x = 1 minute per problem
- 5. 40 math problems means 40 mins
A tank that holds 450 gallons of water can be filled by 1
pipe in 15 minutes, and emptied by another in 30. How long would it
take to fill the tank if both pipes are open?
- 1. 450g / 15mins = 30 gal/min
- 2. 450 / 30 = 15 gal/min
- 3. 30 gal in exceeds 15 gal out by 15 (30-15 = 15)
- 4. 450 / 15 = 30 mins to fill
If a driver completes a trip of 120 miles at a rate of 30mph,
at what rate would the driver have to travel on the return trip in
order to average 40mph for the round trip?
- 1. 120 / 30mph = 4 hours to get there
- 2. Return trip implies same distance back (240 mi total)
- 3. Going 240 mi at 40mph would only take 6 hours roundtrip
- 4. She already used up 4 hours, so she has 2 left to go 120 mi
- 5. 120 mi / 2 hr : 60 mph
A bridge crosses a river that is 1520 feet wide. One bank of the
river holds 1/5 of the bridge and the other holds 1/6. How long is
- 1. 1/5 x + 1/6 x + 1520 = total length (x)
- 2. common denominator is 30, multiply by all
- 3. 6x + 5x + 45600 = 30x
- 4. 45600 / 19 = 2400 ft long
Any question involving gas and gas mileage for hours driving
- 1. Take gallons of gas used * miles per gallon, gives you total miles traveled
- 2. Total miles traveled / average speed = how many hours it took
1 gallon in pints
8 pints = 1 gallon
1 gallon in quarts
4 quarts = 1 gallon
How many lbs in a ton?
2,000 lbs = 1 ton
Total degrees in a shape
(# of sides - 2) * 180
Add to 180
Add to 90
If 2 times r exceeds one-half of t by 5, which of the
following represents the relationship between r and t.
- 2r - (1/2)t = 5
- also same as: 4r - t = 10
One recruiter can complete a certain assignment in 40 minutes;
another can complete the same assignment in one hour. How long would
it take to complete the assignment if they worked together?
- 1. Set equation equal to 1
- 2. x / 40 + x / 60 = 1
- 3. Simplify like the bridge problem using common denom:
- 3x + 2x = 120
- 4. 5x = 120
- 5. x = 24
Questions involving percent discount or reduction
Discount Price / Original Price = % reduction
In order for valve to open twice per secomd, it must rotate at
- 1. 10 spokes/rods on wheel
- 2. twice per second means 120 times per second
- 3. 120 / 10 = 60 rpm
- Increments of .25!
- Think of quarters (change)
- Wheel/Screw/Thread on the right means that turning COUNTERCLOCKWISE (left) tightens, and CLOCKWISE loosens
- Opposite of normal rules
Subtract the number in question from 180
1 nautical mile in ft/meters
- 1 nautical mile: 6,076 ft
- 1 nautical mile: 1,852 m
Time zones are roughly
15 degrees apart in longitude
marks right hand side of a channel when facing inland
Two trains running in the same direction on the same track travel at the rates of 40 and 45 mph, respectively. If the slower train starts out an hour earlier, how long will it take the faster train to catch up with it?
- Distance covered by earlier train / Difference in speed
- 1: 40 miles ahead already / 5 mph difference = 8 hours
Volume of a Cube
- l x w x h
- Cubes also have 12 edges
How much pure acid must be added to 10 ounces of a 55% acid solution in order to produce a 75% acid solution?
- 1. Always start off with 100x due to percentages
- 2. Always multiply first 2 numbers you see
- 3. Other half of equation is Second#(first + x)
- 3. 100x + 550 = 70(10+x)
A farmer who is 6 feet tall wants to determine the height of his barn. He notices that his shadow is 14 feet long and that his barn casts a shadow 70 feet long. How high is the barn?
Set up ratios and cross multiply
If (x – y)2 = 60 and x2 + y2 = 40, then xy =
- Simplify one half and set equal to eachother
- Don't waste a lof of time on it
Jane received grades of 93, 82, and 72 on three tests. What grade must she receive on the next test so that her average for these four tests is 85?
- 1. Add what you have together (call them points)
- 2. Get ideal total points
- 3. Find the difference
A cash box contains a certain number of coins, of which 63 are dimes and 33 are nickels. The rest of the coins in the box are quarters. If the probability of selecting a quarter from this bank is one in five, how many quarters does the bank contain?
- 1. Total # of coins (x) / x + total of what you have
- 2. Set equal to ratio of chance (1/5 in this case)
- 3. Cross multiply
If x2 varies directly as y and if x = 2 when y = 10, what is the value of y when x = 8?
Write out what you see but pay attention to exponents, set up as equation of ratios
Angle E is 40° smaller than its complement. The number of degrees in angle E is
- x + x + your number = 90
- plug in numbers if you have to
Two ships are 2,000 miles apart and sailing toward each other. One sails at the rate of 80 miles per day and the other at the rate of 100 miles per day. How far apart will they be at the end of 9 days?
- 1. first boat distance = rate * time
- 2. second boat distance = rate * time
- 3. Add these and subtract from original distance