Average

Average is calculated by adding a given number of values and then dividing them the number of items. In statistics average is the arithmetic mean and denoted by \rightharpoonup \\X  = \frac{Sum Of elements}{{Number of elements}} = \frac{\Sigma X}{{N}} Example- Average of 3, 5 and 7 is \frac{3+5+7}{{3}}  = \frac{15}{{3}}  = 5 Mean (aka Arithmetic Mean, Average) - The sum of all of the numbers in a list divided by the number of items in that list. For example, the mean of the numbers 2, 3, 7 is 4 since 2+3+7 = 12 and 12 divided by 3 [there are three numbers] is 4.

Average Formula

In working with an average, there is one central formula that is used to answer questions pertaining to an average. This formula can be manipulated in many different ways, enabling test writers to create different iterations on mean problems. The following is the formal mathematical formula for the arithmetic mean (a fancy name for the average). A = \frac{1}{{n}} * \Sigma _{{i=1}}^{n} xi A = average (or arithmetic mean) n = the number of terms (e.g., the number of items or numbers being averaged) x_{1} = the value of each individual item in the list of numbers being averaged The following is the formula for the arithmetic mean, stated in a more readable and understandable form. A = \frac{S}{{N}} A = average (or arithmetic mean) N = the number of terms (e.g., the number of items or numbers being averaged) S = the sum of the numbers in the set of interest (e.g., the sum of the numbers being averaged) One common trap that some students fall into is they automatically divide by 2. However, dividing the sum of numbers by 2 is only correct when there are two terms. When there are more than two terms that are being averaged, dividing by two will give the wrong answer.

Basic Examples

If a teacher tutored five students and they subsequently scored 96, 94, 92, 87, and 81, what was the average score of the students whom the teacher tutored? N = 5 since there are 5 students S = 96 + 94 + 92 + 87 + 81 = 450 A = \frac{S}{N} = \frac{96+94+92+87+81}{5} = \frac{450}{5} = 90 Another Example: If a baseball pitcher throws three straight strikes to the first batter, two strikes to the second batter, one strike to the third batter, and zero strikes to the fourth batter, what is the average number of strikes the pitcher threw to each of the four batters? N = 4 since there are four batters S = 3 + 2 + 1 + 0 = 6 A = \frac{S}{N} = \frac{3+2+1+0}{4} = \frac{6}{4} = 1.5 More Complex Examples A well-respected three point shooter in basketball is shooting 50% from three-point territory (meaning he makes 50% of his three-point shots). If he has attempted 60 three point shots thus far this season, what would his three point percentage be if he made three-fourths of the 12 shots he will attempt during the coming game? N = 60 + 12 = 72 S = 50%(60) + (75%)(12) = 30 + 9 = 39 A = 39/72

Banker's Discount

Definition of banker's discount. The difference between the value shown on a bond, share etc that is bought by a bank from a customer, and the amount that the customer actually receives from the bank. The banker's discount is kept by the bank as payment. Suppose a merchant A buys goods worth, say Rs. 10,000 from another merchant B at a credit of say 5 months. Then, B prepares a bill, called the bill of exchange. A signs this bill and allows B to withdraw the amount from his bank account after exactly 5 months. The date exactly after 5 months is called nominally due date. Three days (known as grace days) are added to it get a date, known as legally due date. Suppose B wants to have the money before the legally due date. Then he can have the money from the banker or a broker, who deducts S.I. on the face vale (i.e., Rs. 10,000 in this case) for the period from the date on which the bill was discounted (i.e., paid by the banker) and the legally due date. This amount is know as Banker's Discount (B.D.). Thus, B.D. is the S.I. on the face value for the period from the date on which the bill was discounted and the legally due date. Banker's Gain (B.G.) = (B.D.) - (T.D.) for the unexpired time. Note: When the date of the bill is not given, grace days are not to be added. IMPORTANT FORMULAE 1.     B.D. = S.I. on bill for unexpired time. 2.    B.G. = (B.D.) - (T.D.) = S.I. on T.D. = \frac{ (T.D.)^{2}}{P.W.} 3.    T.D. P.W. x B.G. 4.    B.D. = [\frac{ Amount \times Rate \times Time}{100}] 5.    T.D. = [\frac{ Amount \times Rate \times Time}{100 + (Rate \times Time)}] 6.    Amount = [\frac{ B.D. \times T.D.}{B.D. - T.D.}] 7.    T.D. = [\frac{ B.G. \times 100}{Rate \times Time}]
1. The banker's discount on a bill due 4 months hence at 15% is Rs. 420. The true discount is: A. Rs. 400 B. Rs. 360 C. Rs. 480 D. Rs. 320 Answer: Option A Explanation: T.D. = \frac{ B.D. \times 100}{100 + (R \times T)} = Rs.  [\frac{ 420 \times 100}{100 + (15 \times \frac{1}{3})}] = Rs. [\frac{ 420 \times 100}{105 }] = Rs. 400. 2. The banker's discount on Rs. 1600 at 15% per annum is the same as true discount on Rs. 1680 for the same time and at the same rate. The time is: A. 3 months B. 4 months C. 6 months D. 8 months Answer: Option B Explanation: S.I. on Rs. 1600 = T.D. on Rs. 1680. Rs. 1600 is the P.W. of Rs. 1680, i.e., Rs. 80 is on Rs. 1600 at 15%. Time = \frac{100 \times 80}{1600 \times 15 }_{{year}} = \frac{1}{3}_{{year}} = 4 months. 3. The banker's gain of a certain sum due 2 years hence at 10% per annum is Rs. 24. The present worth is: A. Rs. 480 B. Rs. 520 C. Rs. 600 D. Rs. 960 Answer: Option C Explanation: T.D. = [\frac{B.G. \times 100}{Rate \times Time}] = Rs. [\frac{24 \times 100}{10 \times 2}] = Rs. 120. P.W. = [\frac{100 \times T.D.}{Rate \times Time}] = Rs. [\frac{100 \times 120}{10\times 2}] = Rs. 600. 4. The banker's discount on a sum of money for  1^{\frac{1}{2}} years is Rs. 558 and the true discount on the same sum for 2 years is Rs. 600. The rate percent is: A. 10% B. 13% C. 12% D. 15% Answer: Option C Explanation: B.D. for  \frac{3}{2} years = Rs. 558. B.D. for 2 years = Rs. [558 \times \frac{2}{{3}}\times 2 ] = Rs. 744 T.D. for 2 years = Rs. 600. Sum = \frac{B.D. \times T.D.}{{B.D. - T.D.}} = Rs. \frac{744 \times 600}{{144}} = Rs. 3100. Thus, Rs. 744 is S.I. on Rs. 3100 for 2 years. Rate = [\frac{100 \times 744}{{3100 \times 2}}]_{{%}} = 12%   5. The banker's gain on a sum due 3 years hence at 12% per annum is Rs. 270. The banker's discount is: A. Rs. 960 B. Rs. 840 C. Rs. 1020 D. Rs. 760 Answer: Option C Explanation: T.D. = [\frac{B.G. \times 100}{{R \times T}}] = Rs. [\frac{270 \times 100}{{12 \times 3}}] = Rs. 750. B.D. = Rs.(750 + 270) = Rs. 1020. 6. The banker's discount of a certain sum of money is Rs. 72 and the true discount on the same sum for the same time is Rs. 60. The sum due is: A. Rs. 360 B. Rs. 432 C. Rs. 540 D. Rs. 1080 Answer: Option A Explanation: Sum = [\frac{B.G. \times T.D.}{{B.D. - T.D.}}] = Rs. [\frac{72 \times 60}{{72 - 60}}] = Rs. [\frac{72 \times 60}{{12}}] = Rs. 360. 7. The certain worth of a certain sum due sometime hence is Rs. 1600 and the true discount is Rs. 160. The banker's gain is: A. Rs. 20 B. Rs. 24 C. Rs. 16 D. Rs. 12 Answer: Option C Explanation: B.G. = \frac{(T.D.)^{2}}{{P.W.}} = Rs. [\frac{160 \times 160}{{1600}}] = Rs. 16.   8. The present worth of a certain bill due sometime hence is Rs. 800 and the true discount is Rs. 36. The banker's discount is: A. Rs. 37 B. Rs. 37.62 C. Rs. 34.38 D. Rs. 38.98 Answer: Option B Explanation: B.G. = \frac{ (T.D.)^{2}}{P.W.} = Rs. (\frac{36 \times 36}{800}) = Rs. 1.62 B.D. = (T.D. + B.G.) = Rs. (36 + 1.62) = Rs. 37.62 (Ans) 9. The banker's gain on a bill due 1 year hence at 12% per annum is Rs. 6. The true discount is: A. Rs. 72 B. Rs. 36 C. Rs. 54 D. Rs. 50 Answer: Option D Explanation: T.D. = (\frac{B.G.\times 100}{R \times T}) = Rs. (\frac{6 \times 100}{12 \times 1}) = Rs. 50.(Ans) 10. The banker's gain on a certain sum due  1_{{2}}^{1} years hence is \frac{3}{25} of the banker's discount. The rate percent is: A.   5_{5}^{1}% B.   9_{11}^{1}% C.    8_{8}^{1}% D.   6_{6}^{1}% Answer: Option B Explanation: Let, B.D = Re. 1. Then, B.G. = Re. \frac{3}{25}. T.D. = (B.D. - B.G.) = Re. (1-\frac{3}{25}) = Re. \frac{22}{25} Sum = [\frac{1 \times (22/25)}{1-(22/25)}] = Rs. [\frac{22}{3}] S.I. on Rs. [\frac{22}{3}] for  1\frac{1}{2} years is Re. 1. Rate =    =  \frac{100}{11} 9\frac{1}{11}% 11. The present worth of a sum due sometime hence is Rs. 576 and the banker's gain is Rs. 16. The true discount is: A. Rs. 36 B. Rs. 72 C. Rs. 48 D. Rs. 96 Answer: Option D Explanation: T.D. = P.W. x B.G. = 576 x 16 = 96. 12. The true discount on a bill of Rs. 540 is Rs. 90. The banker's discount is: A. Rs. 60 B. Rs. 108 C. Rs. 110 D. Rs. 112 Answer: Option B Explanation: P.W. = Rs. (540 - 90) = Rs. 450. S.I. on Rs. 450 = Rs. 90. S.I. on Rs. 540 = Rs.  (\frac{90 \times 540}{450}) = Rs. 108. B.D. = Rs. 108. 13. The banker's discount on a certain sum due 2 years hence is \frac{11}{10} of the true discount. The rate percent is: A. 11% B. 10% C. 5% D. 5.5% Answer: Option C Explanation: Let T.D. be Re. 1. Then, B.D. = Rs. \frac{11}{10} = Rs. 1.10. Sum = Rs. (\frac{1.10\times1}{1.10-1}) = Rs. (\frac{110}{10}) = Rs.  11. S.I. on Rs. 11 for 2 years is Rs. 1.10 Rate = (\frac{100\times1.10}{11\times2})_{{%}} = 5%.