The competitive exam is not testing who is the fastest at long division. It is a test that seeks to measure problem solving skills that are not necessarily the “textbook” ways to discover solutions. (rangerproofswag.com) Let’s discuss some estimation strategies, which are not used as often as they should be.
1. Round Up AND Round Down When Multiplying
Be aware of the direction in which you are altering the result. If you want to estimate a product of two “ugly” numbers, you can move one up and one down, which is an attempt to minimize the error in your estimation. For example:
658*436 = 286,888
If we round 658 UP to 700 and 436 DOWN to 400, we can approximate using:
700*400 = 280,000
2. Round In The Same Direction When Dividing
When you want to approximate a fraction, you can either adjust only the numerator (or denominator) or move both in the same direction. For example:
8/19 = .4210526…
8/20 = 0.4 (Note that increasing the denominator, will decrease the fraction.)
9/20 = 0.45 (Note that increasing both top and bottom will increase the fraction.)
Your estimate is somewhere between .40 and .45.
3. Remember These Other Helpful Tips
Peek at your answer choices: If your answer choices are relatively far apart, this could be hint that approximation is helpful. If the answers are very tight together, you may still estimate, but you have to be more careful and do due diligence.
Geometry shortcut 1: √2 =~ 1.4 and √3 =~ 1.7. Try to commit these to memory, as they are very common.
Geometry shortcut 2: Be careful when using pi = 3. Recognize that you are using a smaller number, so your result will be smaller too. Test makers love to give tempting answer choices that assume pi = 3. It’s not.
Geometry shortcut 3: Even though you cannot assume charts are drawn to scale, they can still be a resource. Obtuse/acute angles are typically shown as much, and angles can be approximated in many circumstances. That’s not to say “if it looks like a right angle, it must be 90.” But you can use the drawing as a guide to your estimation.
Use the extremes: If you are given a range, it helps to plug in those extremes to see between which values your answer falls. This will focus your attention on the cases that are above (or below) those endpoints.
Two Examples
Q 1)If a square has a perimeter of 80 inches, what is the approximate length of its diagonal, in inches?
A. 20
B. 28
C. 40
D. 56
E. 112
This question uses the word “approximate,” so that should be a very big hint that you will need to find a number “close enough.” If P = 80, then s = 20. The diagonal is essentially a hypotenuse of a 45-45-90 triangle, so d = 20√2.
Two strategies:
1) 20√1 = 20 and 20√4 = 40. Therefore 20 < 20√2 < 40. (B) 28 is the only option.
2) Since we remember that √2 =~1.4, we can simply multiply 20*1.4 = 28. (B).
Q 2)Addison High School’s senior class has 160 boys and 200 girls. If 75% of the boys and 84% of the girls plan to attend college, what percentage of the total class plan to attend college?
A. 75
B. 79.5
C. 80
D. 83.5
E. 84
84 is an obscure number. When you see obscure numbers, that is another sign that you may want to look for an approximating shortcut.
Firstly, we should eliminate the overtly incorrect choices. This will be (A) 75 (since that’s the low extreme) and (E) 84 and (D) 83.5 (since they are both essentially equal to the high extreme).
Secondly, find the average of the given percents. Since there are more girls than boys, we know that the weighted average will be closer to the girls’ percent than the boys’ percent. By finding 79.5% as the mean of 75% and 84%, we are given the low extreme. Again, we recognize the weight placed on 84%, making the answer higher than 79.5. (C) 80 it is!
(For similar questions in the future where we actually need to calculate, we could drop the extra “0” from 160 and 200. The ratio of 16:20 is the same (4:5), and the calculation is much easier.)
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