MATH DRILLS Q&A

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  2. Practice Drills – Converting Decimals-Fractions-Pct
    Q10
    If the price of an item is reduced 20% then what additional fractional amount can I purchase?

    First, it is helpful to convert percent into fractions because you can work more easily with reciprocals and % figures can be misleading. A common student mistake is to assume a 20% reduction in price means I can buy 20% more of the item. Not true!
    Second it is important to read what the questions asks. In this case, we want to know the “additional” amount you can buy expressed as a “fraction”. That means the same answer expressed as a “percent” would be the wrong answer. Pay careful attention to what the question wants.

    Explanation:
    The expenditure or total money spent is the price of the item multiplied by the number of items purchased.
    E=PxN expressed algebraically. In simpler terms, you can think of it as I have $1 to spend and I can buy 1 item for the price of $1 per item (a widget is just an item). If the price is reduced 20% then the new price becomes (100% – 20%) = 80% = 4/5 of the old price. New price = 80% x $1 = 4/5 x $1 = $0.80.

    At this point it is worth noting that the first goal is to express the new price as a factor or fractional amount of the old price. We had to take an extra step and subtract the 20% discount in order to derive the new fractional amount, 4/5, to find the new price.

    Now we must figure out… with the same $1 total to spend… how many items can we buy at $0.80 each? (fractional amounts are allowed)
    Expressed algebraically and solving for N, the number of items, our equation becomes N=E/P. That means the total number we can buy will be $1/$0.80 and since we know $0.80 is 4/5 of $1 we can express that amount as…
    $1/(4/5 x $1) = 1/(4/5) …this is the reciprocal of 4/5.
    You can see above, that implies the new amount expressed as a fraction of the old will be the reciprocal of the new price expressed as a fraction of the old price. In this case, the reciprocal of 4/5 is 5/4.
    5/4 can be expressed as a mixed number, 1 1/4. If we can now buy 1 1/4 items and originally we could only buy 1 item then the “additional” “fractional amount” we can purchase is…
    1 1/4 – 1 = 1/4
    1/4 is the added fractional amount and it is the correct answer to the question.

  3. Area-Equilateral Triangles

    I was recently asked if you can calculate the area of an equilateral triangle given only the side-length. The answer is yes! There should be an explanation in the Geometry- polygons instruction unit, but the AREA= (√3 × square of the side-length)/4

    1. Always a great question for any problem Srisai. I believe the drills are randomly delivered so it would help in the future to know what the question is. In this case I believe you are referring to, “If a right triangle has two sides of equal length 2, then the remaining side length must be:”

      There are 3 ways to look at it that may be familiar to you.
      1. If two sides are the same length and it is a right triangle then the hypotenuse forms the diagonal of a square. You may already know that diagonal is S√2 where S = 2 is the square side-length.
      2. Also, you can uses Pythagoras’ theorem. a^2 + b^2 = c^2. In this case, c = √(2^2 + 2^2) = √8 = 2√2
      3. A right isosceles triangle must be a 45-45-90 special right triangle and the hypotenuse is the √2 times the leg lengths.

      I think the last way is fastest and it is very useful to know your special right triangles by heart for the exam to save time and offer possible insight into problem solutions, but feel free to use whichever approach you find easiest.

    1. The current explanation involves shifting decimal places to the left for multiplication and to the right for division by a decimal value. 0.2 (1 digit to the right of the decimal) x 0.2 (1 digit to the right of the decimal) x 0.2 (1 digit to the right of the decimal) = 2 x 2 x 2 = 8.0 then shift the decimal 1 + 1 + 1 = 3 places to the left to get 0.008. Next dividing by 0.4 (1 digit to the right of the decimal) shifts the decimal point one place back to the right after calculating the whole number result. 8.0/4.0 = 2.0 then instead of 3 places to the left, the decimal comes right one decimal and ends up (3 – 1) = 2 places to the left. The result is 0.02.

      Another way to look at it is in terms of powers of 10. 0.2 = 2 x 10^-1. So 0.2 x 0.2 x 0.2 = (2 x 10^-1) x (2 x 10^-1) x (2 x 10^-1). By the commutative and associative property of multiplication we can reorder and regroup this to become (2 x 2 x 2) x (10^-1 x 10^-1 x 10^-1) and noting that you can add exponents when multiplying the same base that becomes 8 x 10^-3 = 0.008.
      Next divide by 4 x 10^-1.
      (8 x 10^-3)/(4 x 10^-1) = (8/4) x ((10^-3)/(10^-1))
      We can subtract exponents when dividing by the same base so this becomes (8/4) x 10^(-3-(-1)) = 2 x 10^(-3+1) = 2 x 10^-2 = 0.02
      You may find the second approach easier.

    1. The grading curve is difficult for the letter grades.;)
      I’m kidding somewhat, but it actually kind of is. I would not worry about the letter grades.
      What is going on numerically is that the score & grade include a deduction for wrong answers to maintain a zero expected value for every problem. The % correct is just that…the % correct/total questions w/o any deduction so the two differ slightly.

    1. Great question Ryan. You will not easily calculate the complete result without or even with a calculator so you have to recognize patterns in the numbers. In a few places on the exam you may be asked to calculate something that appears too complex to reasonably compute without a calculator. Chances are you either need to find a simplification of the expression in terms of some other procedure or value. In this case it is a pattern.

      5^1 = 5
      5^2 = 25
      5^3 = 125
      5^4 = 625
      5^5 = 3125
      5^6 = 15625
      …….

      Note the last 2 digits are “25” for anything greater than 5^1. The hundreds place digit alternates between “1” and “6”. Every even power of 5 has a 6 in the hundreds digit. Every odd power has a 1. 3 displays similar patterns as well. Take a try at that pattern and see what results.

    1. The area of an equilateral triangle is the side length squared times the square root of 3 divided by 4. If S=6 then that is 36 x sqr root 3 / 4 = 9 sqr root 3.

    1. Question 12 is the square root of 100,000. This example is likely included because it is an odd number of zeroes and does not follow the rule exactly. It is not possible to simply divide the number of zeroes in half to get the square root in this case. However, factoring can still be extremely helpful for manipulating the result.

      (100,000)^1/2 = (10 x 10,000)^1/2 = (10)^1/2 x (10,000)^1/2 = 10^1/2 x 100 = 3.16… x 100 ~ 316

  12. How come when I got a 100% on the timed practice, it only says I did better than 67% of test takes? The leader-board says that no one has a better score than 95%. I think the leader-board is glitched. Anyway, is it timed based? Like someone who also got 100% got a faster time? My time was 2:17.

    Thanks for taking the time to read!

    1. Hi maddybrook,

      Your results appear to be correct. Approximately 1/3 of basic math drills are 100% correct, which places you in the top one third. Remember, these are short basic drills. Please try more challenging questions to better differentiate your performance. FYI the leader boards are calculated based on your running average, not individual exams, so nobody who scored 100% always scored 100%. However, the leaders came closer than others.

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