Arithmetic Reasoning Test Improvement

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The biggest issue on this section is discovering what skills they are testing and then to learn those skills. Observe:

! Skills Needed for the Test

* '''Deciding when to add/subtract/multiply/divide'''. If you know which of these steps you need to take with which numbers, then you have solved the problem. Most problems will spell it out, giving you clues to:
** ''subtract'' if the problem says "difference" or "how much more"
** ''add'' if it talks about one number "and" another
** ''divide'' if it asks "how many somethings will fit in some other thing?"
** ''cross-multiply'' if it talks about the "rate" or telling you to "scale" something (such as, a 20 pound bag is $5, how much does 160 pounds cost?). (Another example: "5 inches on a map represents 10 feet. How many feet is something that is 32 inches on the map?")
** ...more examples to come.
* '''Cross-multiplication'''! The ASVAB loves to cross-multiply. So many problems boil down to this. The [Wikipedia article on Cross-multiplication|http://en.wikipedia.org/wiki/Cross-multiplication#Use] is pretty clear. You have to know this.
* '''Understanding that a number can be:''' a percentage (25%), a fraction (1/4 or 4/16), a decimal .25, a ratio (1:4), or "a quarter" and not change its meaning at all.
** To convert a decimal (.25) to a fraction, put the original number over 1 (.25/1) then move the decimal place of both to the right, filling in with zeros, until you get rid of the decimal (25/100), then reduce the fraction (see below).
** To convert a fraction to a decimal, do the long division. 1/4 really means "1 divided by 4".
** To convert a percentage to a decimal, just move the decimal two places to the right. You can think of the % as two zeros (reminding you of how many to move) and an arrow telling you which way to move it (/). So, 25% becomes .25 in decimal.
** To convert a decimal to a percentage just do the opposite---move the decimal place two to the right. So, .25 becomes 25% again.
** Ratios are just fractions with a different symbol. 1:4 is 1/4, or spelled out: "one to four" is "one over four".
* '''Convert units''' (inches/feet/yards, dozens, etc.). Many of the questions add these twists in to boost the difficulty, and most of the time the answer without the conversion is a choice. For example, if the answer is in feet, and it's looking for "3", then "36" is probably a choice.
* '''Filter useless facts out'''. Figure out what information is useful. Some problems have numbers that you don't have to use at all. Just think deep and wide.
* '''"Mess Questions"'''. Some have many numbers and you ''do'' have to use every last one of them. A typical Mess Question involves several conversions of units, many steps, many pieces, and many places to screw up.
* '''Interest (financial)'''. Usually "simple interest", but a few "complex interest" thrown in here and there, but nothing you need to memorize a formula for.
* '''Discounts'''. ASVAB is big on asking, "How much is this after a 25% discount?" and "This was discounted by 3%, how much was it before?" You just need to know: The discounted price is 100% minus the discount (so 75% if the discount is 25%). Multiply the original price by that new percent, and there is the discounted price. Or, to reverse the process, divide by the discounted price. This same trick is useful when asked how many are ''not'' in a group; for example, "32% of 550 people are men. How many people in the group are women?" Just subtract 32% from 100% then multiply: 68% (.68) times 550 is 374.
* '''Price increases'''. The problem, "How much is a $200 product after a 35% price increase?" can easily be solved by adding 100% and 35% then multiplying: $200 times 1.35 is $270.
* '''Time Calculation'''. [Dr. Math has it perfect|http://mathforum.org/library/drmath/view/58426.html]
* '''Distributive Property'''. They want to be sure you know 5 * (10 + 2) is 5*10 + 5*2. I don't think it gets more complicated than that on the test.
* '''Reducing fractions'''. Many times your answer will come out to something like 2/12, but that isn't on the list of choices. So you divide the top (the numerator) ''and'' the bottom (the denominator) by 2, and get: 1/6. When I'm reducing fractions I usually:
** divide by 10 if both numbers end in zero (that is, you drop a zero from the right of both numerator and denominator), 
** divide by 5 if both numbers end in 5
** divide by 2 if the number is even
** try dividing by 3 if the numbers are odd
** try other odd numbers. Once you've divided by everything from 2 to half the number, you can stop. 
** start again from the top any time one of the steps works.

! Other Advice
There are some general tips that will help on any test like this:
* Be sure to double-check the answer for reasonableness. Often you can instantly estimate a range that the answer has to be within. Once you're done with the work make sure the result is in this boundary.
* Work the problem forward and backward. Let's say the problem asks you for 180 minus 13, and you get 167. Then the safe thing to do is try 167 plus 13 and see if it is 180.
* Combine both of the above tricks: Sometimes you can work backwards from the multiple-choice answers. If you pick the best looking one, plug it into the problem, you can find your answer without the heavy work. Sometimes you are close, so you adjust your next guess accordingly.
* Manage your Notes:
** Use clear handwriting---painfully clear. Even if you feel like you are slowing way down, it will save time.
** Clear language with yourself. That is, make sure you can go back and read everything and understand why you wrote it. A good first step is to write the problem number and circle it before you start your work. Also, put a box around the answers. Everything in that region should be just about that one problem. Write down all the units (if something is 6 inches write "6 in", not "6"). Remember that math is already very stripped down and bare, so if you try to do something (like leaving out the "x" symbol in a long multiplication), you are probably making a mess.

! Samples

A few example questions, from [Official-Asvab.com|http://www.official-asvab.com]:
# If the tire of a car rotates at a constant speed of 552 times in one minute, how many times will the tire rotate in half-an-hour?
# One in every 9 people in a town vote for party A. All others vote for party B. How many people vote for party B in a town of 810?
# A motorcycle cost $7,250. If it depreciates by 12% per year, how much will it be worth after one year?
More samples can be found:

* [KapTest.com|http://www.kaptest.com/Military/ASVAB/Practice-ASVAB/ASVAB-Questions/ML_asvab_q2.html] (5 of them)
* [ASVABPrepInfo.com|http://www.asvabprepinfo.com/asvab_arithmetic.htm] (5 of them)

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! Full Notes

This is a new project I'm doing. I'm going to catalog every question on the Peterson's test by the skills needed. Then I'll take the most important skills and make Youtube videos for them. I'll make the process public because the information will have some value even at the early stages.

!! Peterson's Test #1
# take percentage; avoid confusion (negation) - "53% of the 1000 students at Barnett High are girls. How many boys are there in the school?"
# cross-multiply - "If four bananas cost 72¢, how many dozen bananas can be bought for $5.40?"
# cross-multiply (with 1) - "Limes are on sale at a price of 8 limes for $1. How much do two limes cost?"
# distributive property - "Which quantity is NOT equal to 75(32 + 88)?"
# avoid confusion - "A truck with room for 200 bales of hay has room for how many more bales if it is one-half full already?"
# make change (long subtraction) - "Bob buys popcorn for $4.65 using a five-dollar bill. How much change will he receive?"
# cross-multiply (with 1) - "How many 1.5-inch thick boards will fit into a box that is 18 inches tall?"
# cross-multiply - "If potting soil is sold in 20-pound bags for $5.20 each, the price of 160 pounds of potting soil is"
# ignore distractions; reduce fraction; convert units - "A man bought 3 dozen eggs at the store. There are a dozen eggs in each carton. When he got home, he discovered that 2 eggs in each of the cartons were broken. What part of the eggs was not broken?"
# calculate discount - "A jacket costing $96 is discounted by 20 percent. What is the discounted price of the jacket?"
# cross-multiply - "On a house plan on which 2 inches represents 5 feet, the length of a room measures 7 1/2 inches. The actual length of the room, in feet, is"
# calculate discount (compound) - "An appliance store gives a 15% discount off the list price of all of its merchandise. The store gives an additional 30% discount for a floor model. A television set that has a list price of $300 and is a floor model sells for"
# packing - "Craig spent exactly one dollar to buy 3-cent stamps and 5-cent stamps. The number of 5-cent stamps that he could NOT have bought is"
# cross-multiply (with 1) - "A machine punches cards at a rate of 400 per hour. How many hours has the machine been working if it has already punched 1,800 cards?"
# rate arithmetic - "A teacher has 200 tests to grade. If he can grade 40 tests per hour, the number of tests remaining to be graded after he has worked for 3 hours is"
# calculate price increase - "The sticker price of a car is $19,800. If the tax increases the price by 11% and the title increases it by $480, what is the total cost of the car?"
# cross-multiply - "Two inches of track on a scale model railroad represents 20 real miles. On the same model, a distance of 55 real miles is represented by how much track?"
# time arithmetic - "A nurse worked a double shift from 8:45 a.m. Thursday to 3:15 p.m. the next day. If she is paid $20 per hour worked, how much money did she make on this double shift?"
# calculate percentage - "A retailer pays $120 for a stereo and sells it for $195. The retailer's profit is what percent of the price he paid for the stereo?"
# cross-multiply; convert units - "Roy needs to carpet a room that measures 12 feet wide and 15 feet long. What will it cost to carpet the room if carpeting costs $20.80 per square yard?"
# rate arithmetic - "A car that averages 21 miles to the gallon starts a trip with a full 10-gallon tank of gas. If the car travels 150 miles, how many more miles can it go before the tank is empty?"
# profit - "If a man buys lemons at 3 for 30 cents and intends to sell them at 5 for 60 cents, how many lemons does he have to sell in order to make a profit of 50 cents?"
# calculate percentage - "The price of an Allen wrench was reduced from 31 cents to 28 cents. The percent of decrease is approximately"
# unit cost ($) - "Ms. Smith wants to buy 72 ounces of canned corn for the least possible cost. Which of the following should she buy?"
# know ounces in quarts; reduce fractions - "Four fluid ounces is what fractional part of one quart?"
# cross-multiply - "At a price of 99 cents per dozen eggs, the cost of 16 eggs is"
# simple interest; avoid confusion - "A man invests $10,000 in a bond that pays him $5.12 annually for every $100 he invests. How much is he paid at the end of the year?"
# packing - "A plumber needs eight pieces of pipe, each 3 feet 2 inches long. If pipe is sold only in sections of 10 feet each, how many sections must he buy?"
# average two different rates - "A truck driver travels a certain distance at 60 miles per hour and returns over the same road at 40 miles per hour. What is her average rate for the round trip?"
# work equation (cross-multiply) - "Six tractors can plow a cornfield in 8 hours if the farmers all work together. How many hours will it take 4 tractors to do the job?"
!! Peterson's Test #2
# converting fraction to decimal; multiplying decimals - "If a recipe for a cake calls for 2 1/2 cups of flour, and Mary wishes to make three cakes, she must use how many cups of flour?"
# packing; common sense reasoning - "The toll on the Prospect Street Bridge is $1.00 for car and driver and $.75 for each additional passenger. How many people were riding in a car for which the toll was $3.25?"
# time calculation - "Yesterday morning, Janette got up at 7:42 A.M. and went to bed at 10:10 P.M. How much time went by between her getting up and going to bed?"
# calculating discount - "A woman wins a contest prize of $1,000. If 20 percent of the prize is withheld for taxes, the amount the woman actually receives is"
# money math - "A change machine dispensed 25 quarters, 12 dimes, 15 nickels, and 50 pennies. What is the total value of the change dispensed?"
# cross-multiply (with 1) - "A box of 24 nectarines costs $6.00. How much does each nectarine cost?" (good example of with-1, also reverse-plugging and multiply-is-easier-than-divide)
# calculate percent; simple interest - "A company borrows $10,000 from a bank at an annual rate of 8 percent. How much interest will the company owe after one year?"
# avoid confusion; convert units; reduce fractions - "A case of water bottles consists of 4 dozen bottles. If 7 basketball players each drank 6 bottles of water, what part of a dozen was left?"
# geometry (calculate perimeter) - "The side of a square is 20 cm. Find its perimeter."
# multiply; common sense - "Fifteen books with widths of 2 cm each can be arranged on a shelf with no room left over. Find the length of the shelf."
# calculate profit; avoid confusion; money math - "If a muffin is bought at $1.00 and resold at $1.20, find the profit made on 12 dozen muffins."
# convert units (tons to pounds) - "The weight of the load carried by a truck is 15.26 tons. What is the weight of the load in pounds?"
# cross-multiply - "A cake requires two ingredients in the ratio 2:3. If the requirement of the first ingredient is 20 ounces, find the requirement of the second ingredient in ounces."
# geometry (calculate volume) - "Find the volume of a rectangular solid that is 12 meters long, 9 meters wide, and 8 meters high."
# cross-multiply - "In a town, 30% of the members of the population are school-going children. If the number of school-going children is 4,500, find the population of the town."
# calculate percentage - "The local government decides to spend 10% of the budget on education, 30% on healthcare, 20% on municipal facilities, and the remaining money on public transport. If the total budget is $100,000, find the amount spent on public transportation."
# calculate perimeter; money math - "If fencing costs $9.00 per yard, what would be the total cost of the fencing needed to enclose a rectangular area that measures 46 feet by 34 feet?"
# d=rt formula - "Riding a bike to school takes 25 minutes. Coming home only takes 20 minutes. If the trip to school is 3 miles long, what is the average speed, in miles per hour, for the whole trip?"
# cross-multiply; avoid confusion (add $15 to the 80%) - "If a discount of 20% off the marked price of a jacket saves a man $15, how much did he pay for the jacket?"
# cross-multiply - "A graph shows a company's progress toward a sales goal. If 2 inches on the graph represents $10,000 in sales, how much does 19 inches represent in sales?"
# calculate percentage - "Jeanette's salary was increased from $24,000 to $26,000. The percent increase was most nearly"
# algebra - "Judy is three times her daughter's age. If the difference between their ages is 30, how old is the daughter?"
# work formula - "Pipe M can fill a tank in 36 hours, and pipe N can fill it in a certain time. If both pipes are opened simultaneously, the tank gets filled in 24 hours. Find the time it takes pipe N alone to fill the tank."
# simple interest - "A principal of $500, kept at a 4% annual rate of interest, earns $40 as interest. Find the number of years for which the principal was kept."
# geometry; algebra; reasoning - "A farmer owns two rectangular fields. The length of the larger field is twice the length of the smaller field, and the width of the larger field is four times the width of the smaller field. If the smaller field has an area of 1 acre, what is the area of the larger field?"
# money math - "Pencils are bought at 35 cents per dozen. If every lot of 3 pencils is sold for 10 cents, find the total profit on 5.5 dozen pencils."
# divide decimal; ratio/reduce fraction - "The price of a candy bar is $1.00. The price of a pack of a dozen of the same candy bars is $9.60. What is the ratio of the amount saved in buying a dozen pack to the amount spent in buying 12 candy bars individually?"
# reasoning; reduce fraction - "In a group of 15 students, 7 can speak Spanish, 8 can speak French, and 3 can speak neither of the two languages. What fraction of the group can speak both French and Spanish?"
# cross-multiply; convert units - "If 48 feet are represented by 12 inches on a scale drawing, how many feet are represented by 1/4 inch on the drawing?"
# reasoning; work formula - "If four women can wallpaper 2 rooms in 1 day, how many women, working at the same uniform rate, can wallpaper 12 rooms in 2 days?"

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Arithmetic Reasoning Test Improvement