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• Numbers For Mac 10.12 6
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Number series is important for various competitive examinations. In this category of questions, a series of various numbers is given with a blank . We are supposed to find out a pattern between every number and its predecessor and find out the answer using the same logic.

For the purpose of better understanding of the concept, we classify them into the following categories:

The sample variance, s², is used to calculate how varied a sample is. In statistics, a data sample is a set of data collected from a population. It opens in Numbers then I have to save it as an Excel file. That doesn't seem right to me. When I try to open it I get a window for every Excel file on my computer with a message that it can't be opened. Once I close them all I have to open it with Number then save the file as an Excel file.

1. I love using Numbers on my iPhone but I would also love to use it on my MacBook Pro. The problem is I have an older Macbook (2009?) and I have updated it to the latest OS X it can handle (El Capitan 10.11.6) so I can't download the most current Numbers by Apple.
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3. Input: 2, 3/4, 9/12, 3 5/8, -12/16 and order from least to greatest. Convert integers and mixed numbers to improper fractions. 3/4, 9/12 and -12/16 are proper fractions so we can use those as they are written. 2 in fraction form is 2/1. Convert 3 5/8 to an improper fraction. Multiply the whole number 3 by the denominator 8 to get 24.

1. Series with a constant difference
2. Series with an increasing difference
3. Series with a decreasing difference
4. Squares/ Cubes series
5. Combination of different operations
6. Miscellaneous

The best approach :

The best way of approaching number series questions is to observe the difference between various terms. If we see a constant difference, then it’s a constant difference series. If the difference is decreasing or increasing by a constant number , then it is a series of type 2 or type 3. But if there is no such increase or decrease in the difference, then try dividing the 2nd term with the first, third with the second and so on. If you obtain the same number each time, then it is a product series.

Also , if none of these seem to work, then you can try writing each term as a product of two factors and try to see if there is any pattern. If you still observe no pattern and the difference is increasing or decreasing rapidly, then try to look for square/ cube series.

If the difference is increasing and decreasing in some fixed manner, then it is a type of combination series.

Let us look at each type of series in greater depth:

1. Series with a constant difference

In this kind of series, any 2 consecutive numbers have the same difference between them.

For example : 1 , 5 , 9 , 13 , ?

We can observe that we are adding 4 to the previous number to obtain the next number. So, answer here will be 13+4 = 17.

2. Series with an increasing difference

In this type of series, the difference between two consecutive terms keep on increasing as we move forward in a series. Let us try to use this theory in a question.

1,2,4,7,11,16,?

We can clearly observe that the series is increasing with the difference : +1, +2, +3 ,+4 , +5.

So, we will obtain our number by adding 6 to 16 which gives us 22.

3. Series with a decreasing difference

In this type of series, the difference between two consecutive terms keep on decreasing as we move forward in a series. Let us try to use this with some modification in the previous question that we did.

16,11,7,4,2, ?

We can clearly observe that the series is decreasing with the difference : -5, -4, -3 ,-2 .

So, we will obtain our number by subtracting 1 from 2 which gives us 1.

4. Squares/ Cubes series

We can have series where the terms are related to the squares/ cubes of numbers. We can have a lot of variations here. Let us look at some of the possibilities.

1, 9, 25, 49 , ?

We can observe that the above series is square of odd numbers starting from one. So our answer will be 9^2 = 81.

Let us look at another example:

1 , 1 , 2 , 4 , 3 , 9 , 4 , ?

We observe here that the series is formed by writing numbers starting from 1 along with its square as the next number i.e. ( 1 , 1²) , (2, 2²) and so on. So we obtain our answer as 16 which is 4².

Consider the following question:

9 , 28 , ? , 126.

The answer for above question will be 65, let us discuss how.

9 , 28 , ? , 126.

( 2³+1) (3³+1) (5³+1)

The blank should have 4³+1. Hence, the answer is 65.

5. Combination of different operations

This kind of series has more than 1 type of arithmetic operations which have been performed or it can also have 2 different series which have been combined to form a single series. This kind of series is the the most asked and the most important among all the types of series that we have discussed so far.

Consider the series:

1, 3 , 6 , 2 , 6 , 9 , 3 , 9 , ?

The first term 1 is multiplied by 3 to give the second term, 3 has been added to the second term to get the third term. The next term is 2 which is 1 more than the 1st term. It is multiplied with 3 to give next term and the process is continued. With this process, we obtain our answer as 12.

Consider the series:

6, 10 , 7, 11 , 8 , 12 , ?

We can see that the above series is a combination of 2 simple series:

1st , 3rd , 5th terms make an increasing series of 6 , 7 , 8….. . The 2nd , 4th and 7th term make a series of 10 , 11 , 12… . So, our answer will be 9 which is the 7th term of the original series.

6. Miscellaneous series

Some series do not come under any of the above mentioned categories but are very important and also asked in many examinations.

The series of prime numbers or any other related operation done on it comes under this category.

Consider the example:

9, 25 , 49 , 121 , ?

The above series is the squares of prime numbers. So next term will be square of 13 which is 169.

Try out the following questions:

1. 49 , 1625 , 3649 , ?

Solution : Each term in the above series is combination of squares of 2 numbers i.e.

2² 3² , 4² 5² , 6² 7² . So, our answer will be 6481.

2. Look at this series: 40, 40, 26, 26 , 12 , ? … What number should come next?

Solution:Answer is 12. Each number is repeated or firstly 0 is added to each number and then 14 is subtracted from it.

3. 2 , 4 , 11 , 37 , ?

Solution: (2*1) + 2 =4

(4*2) + 3 = 11

(11*3) + 4 =37

(37*4) + 5 = 153

4. 6 , 3 , 3, 4.5 , 9, ?

Numbers For Mac 10.12 6

Solution : We see that no decreasing or increasing difference logic is applicable here. So, we find out the ratios of every term with its predecessor. We get the following values: 0.5 , 1 , 1.5 , 2 . This makes it clear that 9 should be multiplied by 2.5 in order to obtain the next number.

Therefore, the answer is 9* 2.5 = 22.5 .

5. 8 , 15 , 26 , 39 , ?

Solution: Let us start by finding out the difference between every pair of consecutive terms:

15-8=7

26-15=11

39-26=13

We observe that the difference is the series of prime numbers. According to this logic , 17 should be added to 39 to obtain the answer. Hence, the answer is 56.

6. Consider the series: 42, 40, 36, 34, 30, 28, … What number should come next?

Solution: 24 is the answer . we are performing the operations : – 2, -4. )

7. 24 , 30 , 36 , 42 , 54 , 60 , 68 . Find out the wrong term in the series.

Solution: Each term is the sum of 2 consecutive prime numbers.

24 = 11+ 13

30 = 13+ 17

36= 17+ 19

So, according to this logic, 54 is the wrong term. We should have 52 in its place.

8. 17 , ? , 102 , 408 , 2040 . Find out the missing number.

Solution: We can see that the series is increasing rapidly. Let us find out the ratios of every 2 consecutive terms.

2040/ 408 = 5

408/ 102 =4

102/ ? =3

Therefore , our answer is 34.

9. 49, 47, 45 , 42 , 40 ,38 , 35, 33 ,31 , 28, ? , ?

Solution: 26 24 is the answer. It is an alternating subtraction series where 2 is subtracted twice and then 3 is subtracted one and this pattern is continued.

10. 1 , 8 , 9 , 64 , 25 , 216 , ?

Solution: The answer for this is 49. The following pattern is being followed:

1² , 2³ , 3² , 4³ , 5² , 6³ , 7².

11. 1 , 4 , 9 , 1 , 6 , 2 , 5 , ? , ?

Solution: At a first glance it is very difficult to see any kind of pattern here. Any kind of pattern among increasing, decreasing, product , square , cube etc does not seem to be working here.

But we can observe that 1 , 4 , 9 are squares of 1 , 2 , 3 respectively. And after that should come 16 but instead of that we have 1 ,6 . After that instead of 25 , we have 2, 5. So , we can come to the conclusion that when the squares start taking 2 digits, instead of writing them as a 2 digit number , we simply separate them into 2 different terms. So, our next term will be 3 ,6 .

Tip: During the exam, if you feel that you are not able to figure out the pattern in 30-45 seconds, it is better to leave the question for the time being and move on to the next one. You can revisit the question later , if time permits.

CAT Questions related to Logical Reasoning

All questions from CAT Exam Logical Reasoning
Logical Reasoning – Set 1: A high security research lab requires the researchers to set a pass key sequence based on the scan of the five fingers of their left hands.
Logical Reasoning – Set 2: Eight friends: Ajit, Byomkesh, Gargi, Jayanta, Kikira, Manik, Prodosh and Tapesh are going to Delhi from Kolkata
Logical Reasoning – Set 3: In an 8 X 8 chessboard a queen placed anywhere can attack another piece if the piece is present in the same row
Logical Reasoning – Set 4: A tea taster was assigned to rate teas from six different locations – Munnar, Wayanad, Ooty, Darjeeling, Assam and Himachal.
Logical Reasoning – Set 5: Four cars need to travel from Akala (A) to Bakala (B). Two routes are available, one via Mamur (M) and the other via Nanur (N).
Logical Reasoning – Set 6: A new airlines company is planning to start operations in a country.
Logical Reasoning – Set 7: In a square layout of size 5m × 5m, 25 equal sized square platforms of different heights are built.
Logical Reasoning – Set 8: There are 21 employees working in a division, out of whom 10 are special-skilled employees (SE) and the remaining are regular skilled employees (RE).
Logical Reasoning – Set 9: Healthy Bites is a fast food joint serving three items: burgers, fries and ice cream.

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1. Nice article. Categorising number series and then explaining each is very helpful.

2. Such an informative article… Thanks a lot.

3. 367
   564
   Then
   478
   ?
   The answer is not 675

4. □+□+□=30 fill box given below number 1,3,5,7,9,11,13,15 you can repeat any number

5. □+□+□=30
   fill boxes given below numbers
   1,3,5,7,9,11,13,15
   You can also repeat any number

6. 2.3 : 8 :: 3.4 : ?

7. 7, 23, 55,?, 91

8. 8,108,576,2000,5400,? Next number in this series

9. 13 68 42 26 55 ??

10. Complete the series given below. 2,2,2,2,5,9,2,10,__________
    a) 15. C) 28
    D) 23 b)13

11. what comes next 18 11 13 21.5 45

12. 48,55,11? Please tell me the series type and next number in the series ?

13. 0,0,20,115 next no.

14. Given the series:5+3+9+6+13+12+17+24+….Calculate the sum of the first 30 terms of the series.

15. 6,10,8,12,24,60
    Find wrong term

16. 11+13+6=30.

    We can use 9 as six as well.

17. 18:5::12:? Find ratio
    Ans:a )4 b) 10 c)3 d) 6

18. 6,4,15,49?solve this
    A.105
    B.145
    C.188
    D.201

    Tell me this tricky answer

19. 46 21 25 39 54 32
    13 15 19
    9 13 ?

20. 55,100,59,50,?
    Challenge

21. I want to solve some problems

22. Can anyone send answer to this? 2,5,3,8,5,11,6,__??

23. 4 ,25 ,81?

24. Pls tell me soon
    13 ,17,23,33,49,75—?
    Pls solve this series n tell me

25. 6,32,152,450,? Solve it

26. 18,16,10,4,? Solve it

27. 1,1,2,5,8,?,21

28. 49 56 26
    91 72 19
    51 68 ?
    Please solve this

29. 64,32,16,8,_,_ please solve this..

30. 12 5 16
    8 7 18
    6 3 ?

31. 3 ,3,9,11,59,? Ke bad kya hoga

32. 26. 18. 10. 1. 9. 7. 5. 4. 1. 10. 5. ?

33. 3:35 :: 19:? Plz ans

34. 28 7 21
    44 2 ?
    36 4 22
    Please solve this

35. 21,?, 45,180,185
    Options
    42,33 , 41,63

36. 13,17,26,42,67
    options 103,91,109,111

37. 8.5.13.53.276.?

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Equivalent ratios or equal ratios are two ratios that express the same relationship between numbers. The ratio formula and ratio tutorial are provided below the ratio calculator. You can use this equivalent ratio calculator to solve ratio and/or proportion problems by comparing a ratio against an equivalent ratio of the same proportions where the numerator and denominator are a direct multiplication of the multiplying value (m_x). If you need to add, subtract, multiply or divide ratios, use this ratio calculator.

Please rate this Ratio Calculator

From our suite of Ratio Calculators this ratio calculator has the following features:

1. A ratio of 1/2 can be entered into the equivalent ratio calculator as 1:2. 2/10 would be 2:10, 3/4 would be 3:4 and so on
2. The equivalent ratio calculator will produce a table of equivalent ratios which you can print or email to yourself for future reference. You can select how many equivalent ratio examples you need.
3. The equivalent ratio calculator will process decimal ratio values, try entering 2.5:10 into this ratio calculator.

After using the Equivalent Ratio Calculator, other users found these ratio calculators useful:

How to Calculate Equivalent Ratios

As we previously mentioned, Equivalent Ratios are two ratios that express the same relationship between numbers. The Equivalent Ratio Calculator provides a table of equivalent ratios that have the same relationship between each other and directly with the ratio you enter into the calculator. We will look at how to calculate equivalent ratios shortly, first lets look at how to use the free online equivalent ratio calculator:

1. Enter a Ratio into the equivalent ratio calculator, for example, you could enter 7:25
2. Select the number of equivalent ratios that you would like to see in the table of results
3. The equivalent ratio calculator will calculate as you type and produce a lis of equivalent ratios in a table below the calculator
4. [Optional] Print or email the Table of Equivalent Ratios for later use

As we can see, using the equivalent ratio calculator is a quick and easy way to calculate equivalent ratios. This is useful for quick math but you may wish to calculate equivalent ratio manually. That great, understand math formula and calculations is important, particularly ratio math as ratios are used in a significant number of business calculations, finance calculations and general day to day calculations, for example: miles per hour, your BMI, the amount of money your share or sweets you share with friends, these are all good examples of ratios. If you are still working on your mental arithmetic and manual ratio calculations, we suggest that you first enter the ratio into the calculator to see the results, then complete the manual math using the equivalent ratio formula. This way you can check your answers and build up your confidence.

How to Manually Calculate Equivalent Ratios

When calculating equivalent ratios it is important to understand that mathematically, you are expressing the same relationship, simply in different amounts. for example, if you have 10 sweets to share with 4 friends, this is the same and having 5 sweets to share with 2 friends in ratio terms. Although the quantities vary, the ratio of the numerator and denominator are the same (both can be divided by 2 in this instance). The important point to remember is that equivalent literally means equal, so what you are calculating is the same ratio expressed in different quantities. This can only be achieved via multiplication of division.

As equivalent ratios have the same value there is technically no equivalent ratio formula but the following equivalent ratio formula will help you with the manual math calculations.

Equivalent Ratio Formula

We have previously covered the numerator and denominator in the Fractions Calculator, if you are not familiar with the numerator and denominator or simply wish to refresh your memory please review that article and supporting calculator before proceeding.

where

• n = numerator
• d = denominator
• a = multiplier

In our equivalent ratio formula, we can see that by multiplying both the numerator and denominator by the same amount (a) that we maintain the relationship with all equivalent ratio and our initial ratio from which we started the calculation

More Good Ratio Calculators

If you found the Equivalent Ratio Calculator, you will probably find the following ratio calculators useful:

What is a ratio?

A ratio is a direct comparison of one number against another. A ratio calculator looks to define the relationship that compares between those two numbers

Where are Ratio Calculations Used?

Ratios are used everywhere, from cooking with your favourite recipes to building housing, here are some common applications of ratios in everyday life:

• Mechanics: spanners/wrenches are all marked with relevant ratios which correspond to the nuts they fit.
• Businesses and accountants: use ratios for forecasting and financial controls (expense ratio, turnover ratio, debt ratio, asset ratio, price ratio, earnings ratio etc.)
• Food: the correct diet has the right ratio of food groups (one of our five a day is a common ratio we are all familiar with.
• Education: ensuring the right ratio of students to teachers is key for effective learning. Class sizes in terms of the ratio of pupils to a teacher is a common ratio concern.
• Web Developers / SEO experts: these tech guys live and breathe ratios, from bounce rates to time on site, new visitors versus return visitors, ratios rule their lives.

How to Calculate Ratios

When calculating equivalent ratios you must multiply or divide both numbers in the ratio. This keeps both numbers in direct relation to each other. So, a ratio of 2/3 has an equivalent ratio of 4/6: in this ratio calculation we simply multiplied both 2 and 3 by 2.

How To Update Mac 10.12.6

Math Calculators

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