Friday, September 6, 2013

Counting & Venn Diagram: Solved Example(Counting, making a systematic list, using arithmetic to count, the counting principle, venn diagram)

Question: 1 From 1:13 pm John read pages 109 to 113 of a novel. What is his rate of reading in pages per minute?
(a) 3/5  (b) 3/4  (c) 4/5  (d) 1  (e) 5/4

Explore: John read pages 103 to 113. Both page 109 & 113 are included. So, he read 113-109 +1=4+1=5 pages. He3 read form 1:09 to 1:13 pm. Here the 9th minute is not included. So, he read for 13-9=4 minutes.

Answer: (e) 5/4  

Question: 2 How many integers are there between 100 & 1000 all of whose digits are odd?

Explore: Since the number is between 100 & 1000, it is obviously a 3- digit number like 135, 533, 759 etc. in which all three digits are odd. Certainly, it is not feasible to list all the numbers. But we can count them by listing a few of them in an order fashion and 115, 117, 119, 121 cannot be in the sequence since 2 is even. We will again start from 131 & go on like 133, 135, 137, 139. So, we have a pattern between 110 & 120, we have 5 numbers; between 130-140 we have 5 numbers, & son. So, in the range 100-200 there are 5 such intervals each containing 5 numbers giving a total of 5x5=25 numbers.
Now within the range 200-300, none of the number meets our criteria,, since all of them start with 2, an even digit.

Within the range 300-400 we again find 25 numbers meeting our criterion. There are 5 such intervals (100-200, 300-400, 500-600, 700-800, 900-1000). So, the total number of required integers = 5x25 = 125.

Alternative way

The best way to solve the above problem is by using the counting principle. Note that we have 5 odd digits: 1, 3, 5, 7, 9. We want to make 3-digit integers. This task may be fragmented into 3 tasks:
(1) Select the 1th digit (100s place)
(2) Select the 2th digit (10s place)
(3) Select the 3th digit (unit place)

Since there are 5 odd digits, we can do task 1 in 5 ways. Similarly, tasks 2 & 3 can also be done in 5 ways each. So, the total number of ways = 5x5x5= 125.

Question: 3 In how many ways can you arrange the letters A, B, C & D?
(A)2  (B)6  (C)8  (D)13  (E)24 

Explore: Let us think of arranging the letters in each position as a task & use the counting pricniple. The 1st letter may be chosen in 4 ways. It can be any one of  A, B, C or D. Having used one letter, the 2th letter may be chosen in 3 ways. The 3rd in 2 ways & the last in only 1 way. So, the number of possible arrangements = 4x3x2x1=24.

Answer: (e)24

Question: 4 A survey of the town of Mel-bond found that 70% of the people watched the news on TV, 35% of the people read the news on newspaper & watched the TV and 25% read a newspaper & watched the TV news. What percentage of the people either watched the news on TV nor read it on the newspaper? 
(A)30%  (B)0%  (C)20%  (D)15%  (E)5%

Explore: The following Venn diagram will help to visualize the situation:
Let, x be the required percentage. So, we can write 45 + 25 + 10 + x = 100[ Total 100%]
Or, x = 100-80 = 20%

Answer: (C) 20%

Question: 5 According to a survey of business firms in a certain city, 750 firms offer their employees health insurance, 640 offer dental insurance, and 280 offer health insurance and dental insurance. How many firms offer their employees health insurance or dental insurance?

Explore: this problem can easily be solved by using Venn diagram.

:. 470 firms offer only health insurance.
360 firms offer only dental insurance.
280 firms offer both health & dental insurance
:.(470 + 360+ 280) or 1110 firms offer health or dental insurance.

Question: 6 Of the 65 families in a locality, 45 have children and 10 have retired individuals as members. Of the families, 8 have both children and retired individuals as members. How many of the families have neither children nor retired in individuals as members ?


:. No. of families have neither children nor retired individuals
= 65-(37 + 8+ 2)
Answer: 18

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