Arithmetic

Ex

.35 + .78 can also be written as .35 + .78 =1.13(*) .69 - .14 = .55 (Deci. points are aligned, and in the sum, the whole # is placed to the left of the deci. point*)

EX of when the denominators are not the same

1: Find common denominator, or LCD(use multiplication to identify LCD) 2:1/3 + 1/2 + 3/4 = ? (3,6,9,12/2,4,,8,10,12/4,8,12) 3:LCD is 12. Thus, divide original denominators(3,2,4) of each fraction into the common deno. ex. 12/3=4; 4x1=4(following rule of previous fraction)= 4/12. The others would be 6/12, 9/12. 4: Then add the numer. together 4/12 + 6/12 + 9/12 = 19/12(m# becomes 1 7/12)

EX

3/7 x 2/4 = 6/28(reduced to lowest term, 3/14*)

Ex

5/8 + 6/8 = 11/8(M# is 1 3/8*) 4/5 - 1/5 = 3/5

Ex

5/8 / 3/4 = 5/8 x 4/3 = 20/24. (Reduced to its lowest terms, this becomes 5/6*).

Ex

6.021 x .4 → 6021 x 4 = 24084(2.4084) There are 3 DP places in 6.021 and 1 in .4. 3+1=4, so that is how many DP are needed in the result.

Basic formulas and measures

60 seconds = 1 minute; 60 minutes = 1 hour; 24 hours = 1 day Area = L x W (the area of a square or rectangle is obtained by multiplying its length by its width) Rate = Distance divided by Time (R = D/T) Time = Distance divided by Rate (T = D/R) Distance = Rate multiplied by Time (D = R x T)

How to calculate a mixed #

9 1/4 multiplying the denominator of the fraction by the whole number and adding this number to the numerator -- [(4 x 9) + 1]/4 = 37/4. This says that 9 1/4 is the same as 37/4.

Adding and Subtracting Decimals

A decimal is a fraction that is expressed in another form. Numbers that begin with a period (called a decimal point) are decimals (also called decimal fractions).

EX

A passenger purchased a vase from an art dealer in Europe for $1,210. She later learned that the dealer sold her the vase for 110% of its actual value. What was the actual value of the vase? First, think carefully about what you need to find in order to solve the problem. In this problem, you want to find the actual value of the vase (the unknown value). Do this by substituting X for the actual value. Since the vase was sold for $1,210 and $1,210 is 110% of its actual value, the equation should look like the following: $1,210 = 110% of X; this is also written as $1,210 = 1.10X This is also the same as 1.10X = 1,210. Move your known values to the right side of the equation by dividing each side of the equation by 1.10: 1.10X / 1.10 = 1,210 / 1.10. The equation then becomes X = 1,210 / 1.10, which is equal to 1,100 or $1,100. Therefore, the actual value of the vase is $1,100.

Percentage

A percentage is yet another way of expressing a fraction. Writing 100% is the same as writing the fraction 100/100, and writing 23% is the same as 23/100.

Ratio and proportions

A ratio expresses the relationship in quantity, amount, or size between two or more things.

Adding and Subtracting Fractions

Adding & subtracting fractions that have same denominator then add or sub the numerators and retain the denominator

Ex

An agency purchased surplus computer printers priced at $220 each. For every 20 printers purchased, the twentieth printer was purchased at a 40% discount. What equation represents the total price paid, if the agency purchased 100 of the printers? 100 (220) - [ 100/20 x 220 x .40 ] The total price of the computers, without the discount, is represented by 100 (220) = $22,000. The number of printers that were purchased at a discount (every 20th computer) is represented by 100/20, which equals 5. So, the cost of 5 printers priced at $220 is represented by 100/20 x 220. A 40% discount for this price is represented by 100/20 x 220 x .40, the results of which must be subtracted from $22,000 to obtain the total discounted price paid by the agency.

Ex

An importer under investigation sent 500 containers to the United States. Two hundred containers were shipped to New Jersey, and the remaining containers were shipped to Pennsylvania and Virginia in the ratio of 2:3. How many of the containers were shipped to Pennsylvania and how many were shipped to Virginia? The ratio of 2 to 3 tells us that out of every 5 containers, 2 were shipped to Pennsylvania and 3 were shipped to Virginia. Two out of 5 has the same meaning as the fraction 2/5 or the decimal quantity .40; 3 of 5 is the same as the fraction 3/5 or .60. Since we are told that 200 of the 500 containers were shipped to New Jersey, this leaves 300 containers that were shipped either to Pennsylvania or to Virginia. To determine how many containers were shipped to Pennsylvania, we set up the proportion 2/5 = X/300. Set up the problem as such and cross multiply; 5x=300*2=600 5x/5=600/5 x=120 Check, 300-120=180(shipments to VA)

Ex2

An officer rented a car for six days and was charged $450. The car rental company charged $35 per day plus $.30 per mile driven. How many miles did the officer drive the car? The answer can be obtained by letting X represent the number of miles driven and computing the following: 6 (35) + .30X = 450 210 + .30X = 450; .30X = 450 - 210; .30X = 240; X = 240 / .30 = 800 (The officer drove 800 miles). The officer rented the car for six days at $35 per day, which is $210; $210 subtracted from the total charge of $450 leaves $240, the portion of the total charge expended for the miles driven. This amount divided by the charge per mile ($240/.30) gives the number of miles (800) driven by the officer.

Calculating %

CBP Officer Crawford inspected 400 containers for undeclared goods. If 18% of the containers held undeclared goods, how many containers held undeclared goods? 400 x .18 = 72. So, 72 containers held undeclared goods.

Word problems using decimals

Cargo weighing 6,520 tons arrived at the Marin Port and was assessed a fee of 6 cents per ton. What was the total amount assessed on the cargo?(they are asking for total. Multiply) 6 cents = $.06; 6,520 x .06 = 391.20. So, the answer is $391.20. If inspection stickers cost 30 cents each, how many stickers can be purchased for $12.60? (the question is asking for how many, divide)/*ask kelsey if this is true

Solving for Distance, Time and Rate problems

Distance(miles), time, rate(speed) Rate = Distance divided by Time (R = D/T) Time = Distance divided by Rate (T = D/R) Distance = Rate multiplied by Time (D = R x T)

Dividing Decimals

Dividing decimals also requires you to count the number of decimal places.

Calculating % increase

Each year an office was allocated funds to provide bonuses to all of its employees. One year, the office received the same amount of bonus funds, but lost 10% of its staff. By what percentage will the bonus amounts increase for the remaining staff in that office? One way to solve the problem is to let 100% represent the total staff in the office. 100% - 10% = 90%. If the bonus funds previously given to the 10% of employees who left were divided among those who remained, you would get .10 .90 = .111 (11.1% or rounded to 11%). So, the bonus amounts for the remaining staff would increase by 11%.

EX

If you want to know the probability of randomly pulling a lemon bar from a bag of pastries that contains 3 lemon bars, 5 glazed doughnuts, and 2 muffins, you would first need to consider the total number of pastries that are in the bag (3 + 5 + 2 = 10). This total number of pastries in the bag is known as the "sample space." The probability of pulling a lemon bar from the bag would be 3/10 or .30 or 30%. The probability of pulling a glazed doughnut from the bag would be 5/10 or .50 or 50%, and the probability of pulling a muffin from the bag would be 2/10 or .20 or 20%. Should you need another lemon, the equation would reduce to such; 2/9.

EX

In calculating 2.64 /.02, it will be easier to first move the decimal point 2 places to the right in each set of numbers. This will give you 264 / 2, which is equal to 132. Check: multiplying 132 x .02 = 2.64

Calculating % increase/decrease

Increase: In June 2003, the number of CBP Agriculture Specialists employed full-time in one large office was 80. One year later, that number increased by 15%. What was the total number of CBP Agriculture Specialists employed in the office in June 2004? There are two ways to solve this problem. One way is to multiply 80 x .15 = 12; then add 12 + 80 = 92. Decrease: Had the officer workforce decreased 15% from June 2003 to June 2004, the calculation would have been 80 x .85 = 68. (Note that the decimal .85 is used because 100% minus 15% equals 85%, expressed as .85 in decimal form.) This is the same as multiplying 80 x .15 = 12 and subtracting 80 - 12 = 68.

Multiplying fraction

Multiply numerators, then multiply denominators

Expressing Word problems as Equations*

Some of the word problems in the test require you to choose, from among several equations, the one equation that expresses or represents a solution to the problem.

Ex2

The average weight of 3 canines working at a medium-sized airport is 130 lbs. and the smallest canine weighs 110 lbs. If the other two canines are of equal weight, how much does each of the two canines weigh? To solve the problem, first find the total weight in pounds by multiplying 3 x 130 lbs. = 390. Then, subtract the weight of the smallest canine, 390 - 110, to get the combined weight of the two remaining canines, which is 280. Since the two remaining canines weigh the same, divide the result by 2: 280 2 = 140. Therefore, the two remaining canines each weigh 140 lbs. Check: The average of 110 + 140 + 140 = 390 3 = 130 lbs.

Denominator

The bottom # in the fraction

Numerator

The top # in the fraction

Ex

To find the average of 4, 8, and 15, first find the sum of the numbers (4 + 8 + 15 = 27); and then divide the sum by 3 (27 / 3 = 9).

Solving for unknown values

To solve a math problem that has one or more unknown (not given) values, set up an equation to represent all of the values in the problem, substituting a letter of the alphabet, such as a, b, x, or y, for the unknown value.

Multiplying Decimals

You do not need to align DP. Count the # of the deci. places(to the right of the deci. point) in each # and add the two counts together. After multiplying, place the deci. point at the total # of places you counted.

Calculating Averages

add the set of numbers and divide the result by the number of items in the set:

Probability

any number of events occur in our environment. An event can be the tossing of a coin, the selection of an object from a group of objects, the occurrence of a test score among a number of test scores, etc.

Find a % of a number

change the percent to a decimal and multiply by the decimal: 16% of 40 = 40 x .16 = 6.40

Dividing fractions

invert the second fraction (the divisor), and multiply the numerators and denominators.

Fraction

is a number that represents a part of a whole number. A fraction is a division or ratio of two whole numbers,

Whole #

is an integer (0, 1, 2, 3, 4, 5, ...) which can be divided by itself and by 1

Mixed #

is the combination of a whole number and a fraction.

Changing a % ti a decimal,

move the decimal point two places to the left and drop the percent sign: 57% = 57.0% = .57

Calculate what % one number is of another number

reverse the calculation above and divide: What percent of 40 is 6.40 (also expressed as 6.40 is what percent of 40)? Calculate 6.40 / 40 = .16; move the decimal 2 places to the right in .16 to get 16%.