Don’t hate math because it’s intricate and logical. You can love this addition to your brain power. You can use math to startle yourself and others by solving separate, related puzzles. Life issues demand our attention to relationships--not just social relationships.
Just about every decision, whether deciding to stay up longer to watch a movie or deciding which used car to buy, will affect the outcome of other decisions. How does the information from one medical report relate to another? What does the tax form indicate when the "credit" is more than the tax? Oftentimes, we solve several problems before even tackling a larger one.
The Feb. 26 math question requires the solution of small problems to get the answer to the larger question. This briefly stated problem and solution encloses several tips in a very short space. When the question and answer is short, like this one, my explanation can help you to put this pattern in your head to use in other ways. The explanation, in words, tends to look long (I know), but once you follow the visual in your head of the longer explanation, the short problem and answer that I choose for these steps will make a pattern to replace all the words when you see this style of question on a test. You can solve it, but first you have to recognize the real question you are asked.
As I have suggested before, you will be able to use this technique better if you pretend you are walking with me and listening with visualization when I do this particular strategy of choosing a short Q&A to go through as though you have never seen one like this before. The outstanding benefit is due to the visual factor of learning the critical thinking skills from a good, short pattern.
Once you follow the steps of the pattern carefully, noting what your brain is doing while making the connections, you can move through the short pattern efficiently.
The seemingly long explanation becomes a short, visual pattern of critical thinking in your brain. You can use this understanding as a pattern to notice other logical steps in other problems, including non-mathematical problems.
As always, what I am asking you to do is to notice the logic of an alternative way of connecting to answer the problem. The more you do this, the more you will handle aptitude questions, such as those on the SAT with confidence. The question may not look familiar at first, but you will know that your reading of the question and array of answers may still yield the correct answer to you based upon other things you do know.
Words to add to your glossary searches, crossword puzzles, Powerpoint cards, and games:
sum, product, mean (median, mode) and integer.
Here is today’s (Feb. 26) SAT question:
The sum, product, and average (arithmetic mean) of three integers are equal. If two of the integers are 0 and -5, the third integer is
A) -5
B) 0
C) 2
D) 5
E) 10
You have three integers, 0,-5, and x, whose sum, product, and average are equal. Since 0 is one of the integers, the product is 0. So the sum and average are also equal to 0 (zero).
Therefore, -5+x+0=0 So x=5-0 x=5
Know the Words:
Know these words with a quick flash image in your mind: sum,product,mean,integer
Note: I am using ‘I’ rather than ‘x’ for the unknown integer.
Here is what you know: The three integers together result in the same number when multiplied together, added together, and averaged together. (To average the numbers, you add them and then divide that sum by the number of numbers. This is the mean.)
I am calling the unknown integer “I”
So far, you know this much about each problem concerning addition, multiplication, and average of the three numbers. You know what results from the two integers you are given:
0+(-5)=sum=-5
0(-5)=product=0
0+(-5)+i=sum of the three integers
0-5+i=___
I= -0+5=5 i=5 Answer is D
Remember the question! Be careful not to answer another question instead. This is an ongoing strategy for standardized tests in general and the SAT test in particular.
SAT test questions are not designed to trick you. The questions are designed to identify students who will do well in today’s continuing education—into college and beyond.
Tell me again. What is this question?
You are looking for the third integer, not the equal answer. You are testing the ways to find the sum, the product, and the mean because the answer to these three questions is all the same (equal). If by using what you know, you can find the answer to ONE of the three problems that yield the same answer, you will also know the other two answers. Notice that this answer is NOT the answer to the question, just the answer to what you need to get that answer.
What was the question?
New information emerges about the unknown integer because of the relationship of the integers. Keep looking back to remember the directions and the question. Use scratch paper and put a dot in the answer you choose on the test until you have some time to fill these in, according to your pacing. Never go longer than two questions before filling in the scan sheet.
Check: Remember, you are asked to find the third integer. You are told all three answers TO ANOTHER QUESTION are equal. You are not asked to find the equal answer to each problem. You use that number to find the number you need. You are asked to find the third integer.
Notice that the answer possibility ‘0’ IS one of the choices. Don’t go there! Step back from the edge!!!!
This problem is a good pattern to pay attention, to check back, and to prove it.
The sum of the three integers (0,-5,--and, now, +5)
Notice I put that plus in there. Stay focused with some plus signs! Okay, here are the three i-guys (my shorthand for integers): 0,5,-5 in the three problems with the same answer. Add them, multiply them, or add them and then divide by the number of integers (3). The answers are zero, zero, and zero. WAIT! The zero answers are the answers to the three different little problems with equal results.
YOUR QUESTION WAS (look back!): the third integer is _____. The third integer is 5 (D).
0 + 5-5=0 sum
0(-5)(+5)=0 product
0+5+(-5)=0 divided by 3 integers to get the mean 0 divided by 3 = 0 mean
MORE STRATEGIES:
Write this question, answer, and solution on a card. Notice the ways the question tests your aptitude.
Keep on noticing words in every type of book—from baby books to college textbooks.
Sum =the answer to an addition problem. See an image in your mind.
Product=the answer to a multiplication problem.
On your scratch sheet, make a number line with zero in the middle and a few numbers to the left (minus or negative numbers) and a few numbers to the right (plus or positive numbers): Integers are whole numbers and zero.
Enough to know about integers for now: What are integers? Integers are whole numbers to the left and right of zero and zero.
Negative whole numbers are integers. Positive whole numbers are integers. Zero is an integer. Numbers broken into decimals or fractions are NOT integers.
Finding the mean is a quick two-step: Mean is the “average” of two or more integers. First, you add the numbers; then, you divide the sum by the number of integers.
The easiest way to remember how to find the mean is to remember how to find your average for a week of quizzes .
Picture this: You have 10 quiz grades. To calculate the mean, the average, you add all ten grades and then divide the sum (see what I did there?) by 10, the number of quiz grades (integers).
Let’s just say you made a hundred on all but two. On those two, you made 80 on one and 20 on the other one (!). 100 +100+100+100+100+100+100+100+80+20=900
Your average is 900 total points (the sum) divided by 10 grades. You have a 90 average.
Now, look what would happen if you only had five grades, and three were 100; then, you had that 80 and a 20. The total points (sum) would be 400 divided by 5 grades. The average will be 80. Those high grades still helped you to stay with a B.
What if the teacher gives only two grades ? You make 100 on one and 20 on the other. You have 120 points divided over two grades. Your average (mean) is 60. Is that mean? Some students think so.
You have three integers, -5, 0, and x (the unknown—if it helps, label it ‘I for integer’ in your scratch notes).
What you know: There are three integers. You know that one is -5 and one is 0.
The third integer is unknown (i).
If you add the three integers together (-5+0+i), you get -5 + I (the sum).
If you multiply the three integers together (-5x0xi), you get 0 (the product) (Notice, you have part of the answer already right here because when you multiply -5 x 0, anything else multiplied by this 0 will be 0 (zero).
Please keep your wits about you on these little questions with small numbers. You can find the answer, logically, even if you do not know the most efficient way to do it. But you do not have time to hang around a question like this just because you know part of it.
THE MOST IMPORTANT THING IS TO LOOK BACK AT THE QUESTION. YOU ARE LOOKING FOR THE OTHER INTEGER NOT THE ANSWER THAT IS THE SAME EACH WAY THE PROBLEM IS DONE. THAT ANSWER, WHICH IS ZERO, leads you to the answer to the question asked. Notice it is also given as one of the answer choices.
You are looking for the number that results in zero as the answer when the three integers are added together, when they are multiplied together, and when they are averaged together.
You have three integers, 0,-5, and x, whose sum, product, and average are equal. Since 0 is one of the integers, the product is 0. So the sum and average are also equal to 0 (zero).
Therefore, -5+x+0=0 So x=5-0 x=5
The answer is 5. The answer is D.
So check it in the last one by substituting 0 into the answer.
NOTE: This is where you can go astray if you do not know, immediately, how to make a little model. What you know at this point iS NOT THE OTHER INTEGER. IT IS THE ANSWER THAT YOU GET WHEN YOU PUT THAT INTEGER IN THE PROBLEM AS THE THIRD INTEGER (AGAIN, NOT the integer but the sum, the product, the average of the three integers are all the same (all equal).
If you add the three integers together (-5+0+i) or (-5+0+5) = 0 and divide this total by the number of integers (3), you get 0.
The third integer is 5. The answer to each of the three problems is zero every time.
The way to get the 5? You know everything else, so you find 5 as ‘I’ or ‘x’
-5+0+I/0 = 0
I=5 because -5+0+5 divided by 0 is zero.
