Number sense relates to our understanding of numbers and ability to manipulate them. It is because of our number sense skills that makes us aware of how big a number is in terms of units or any other quantity. Number sense is one of the reason behind our ability to compare two numbers or mathematical factors.

Note that the definition of number sense might be different for individuals in different age groups and grades. For kindergarteners, it could be the question how big the fort of blocks is, and for high schoolers, it might be the area covered within two line equations. All in all, number sense is a serious skill to be good at if you want to pick up a good pace in the subject.

Intervention is very much required for students struggling to develop number sense. Certain measures adapted at an early period can be really beneficial. In general, number sense is developed over time as we move to higher classes. Dealing with numbers on regular basis is one such factors behind. Also, as we deal with the same elements on daily basis, majorly we develop a short version in our mind to get the solution out of it in a quick second.

Keeping this in mind, Today we are writing this post on 8 really helpful number sense tricks that will saving a lot of time in your math practice. Few points before we jump to the topic, the tricks given here involves the topics that are used generally throughout our time in school and not just in specific class. Higher level number sense tricks are not included, as that is the focus of another upcoming post. It is recommended that you make a note of it or write it down to revise it on regular basis to really get a good hold of it.

1.) **Multiplication of milestones**

If you do visit our site regularly, then you probably know about what we call ‘milestones numbers’. 25, 50, 75 and 100 are the milestone numbers. Multiplying any number with the milestone is relatively easier. All you need is that when multiplying with milestones, consider turning milestone into a fraction and multiplying the same with 100. We have,

**25 = (1/4) x 100**

**50 = (1/2) x 100**

**75 = 25 x 3 = (3 x 1/4) x 100 = (3/4) x 100**

Taking ’25’ and ’50’ as the pivot point, you can convert further milestones as well, such as **125 (5/4 x 100), 150 (3/2 x 100), 175 (7/5 x 100)** and so on.

2.) **Specific multiplication operations**

Milestone method couldn’t be applied to every other number as it will make the problem more complicated and would actually defeat the purpose. However, specific tricks can be applied to multiplying to certain numbers. Some of them are :

**Multiplication with 11**: For 2 digit (say, 54)-> The resultant will be in this form (First digit ‘5’, Sum of digits’5+4′, last digit ‘4’). The answer is**594****Multiplication with 11**: For 3 digit (say, 138)->The resultant will be in this form (First digit ‘1’, Sum of 1st and 2nd digits’1+3′, Sum of 2nd and 3rd digit ‘3 + 8’ (Don’t forget to add the carry number 1 to the left), last digit ‘8’). We get (1, 4, (1)1, 8). After adding the carry number, we get the answer i.e**1518**.**Multiplication with 12**: Multiply the number by 10 and 2, making it in the form**a x (10 + 2)**. Add both. The resultant is your answer. Example: 12×**7**=**70**+**14**= 84**Multiplication with 15**: Multiply the number by 10, then add half again. Example: 15×**8**=**80**+**40**= 120

3.) **Operations of reverses**

There is a very simple trick to find the difference of a three number from its reverse. In this, you have to

- Subtract the first and last digit/pair
- Then multiply the difference by 100
- Subtract the difference from the resultant

Let us solve:

-> 653 – 356 = **297**

Doing it by the trick,

- (6 – 3) = 3
- 3 x 100 = 300
- 300 – 3 =
**297**

It also works when the resultant difference is a negative number but you need to careful

Let us do the example of **345 – 543**,

- (3 – 5) = -2
- -2 x 100 = -200
- -200 – (-2) =
**-198**

4.) **Difference of squares**

In case of finding difference of squares, it is best to use the expression a² – b² = (a+b)(a-b). It is pretty basic trick and way easier than finding individual squares and then finding the difference. Let us do few examples:

1 ->(52)²-(48)² = **(52-48) (52+48)**

= (4)(100) = **400**

2 -> (123)² – (103)² = **(123 + 103)(123 – 103)**

= (227) (20) = **4540**

5.) **Division series **

Doing calculations in an exam, it is always helpful to know if a number is divisible by the divisor by just looking at it. You don’t want to caught yourself in lengthy calculations and still not able to find the answer you are looking for.

There is no single way to tell if the number is divisible with the given divisor. The method is different for different divisor number. Let us learn about some of the most common ones.

**2**: Check if the number is**even****3**: Check if the**sum**of the all the digits of the dividend is divisible by**3****4**: Check if the number made of**last two digits**of the dividend is divisible by**4****5**: Check if the dividend number**ends**at**‘0’ or ‘5’**.**7**: Take the**last digit, double it and subtract it**from the rest of the number. If the given number is divisible by**7**.**8**: Check if the number made of**last three digits**of the dividend is divisible by**8****9**: Check if the**sum**of the all the digits of the dividend is divisible by**9****11**: Check if the difference of sum of alternate digits of the dividend number is either ‘‘ or divisible by ‘**11**‘

6.) **Square roots and cube roots operations**

Some may find working with square roots and cube roots really difficult, specially when they need to solved in an mathematical expression. Usually in the case of multiplication and division problems involving roots, it the best to simplify to an extent where we can possibly take out the bracket roots. Let us solve this expression:

-> **(∛3 x √3** x

**√5 x ∛9 x √27)/ (√125)**

Here, it is next to impossible to solve by finding individual roots and doing the operations. Instead, we will look possible operations that can be performed without taking out the roots bracket and arriving at numbers whose roots are way easier to find. Let’s solve

Now, if we multiply **∛3** to the **∛9**, we get **∛27, **and 27 square root,as we know, is 3. Also, dividing **√125** with **√5**, we get **√**25 in the denominator. 5 is the square root of 25, so we will write it down as well. The expression becomes

-> **(∛27 x √3 x √27)/(√25)**

-> **(3 x √3 x √27)/5**

-> **(3 x √81)/5**

-> **(3 x 9)/5** => **27/5**

7.) **Fraction to remember**

Math is all about applying logic and solving the problem. However, when you have to solve lengthy questions in a tight time-frame, its better to memorize certain parts in the topic, specially formulas and direct relations. Do note that we are not recommending to put all of it in your mind right away, but develop a habit of memorizing certain parts while practicing so that you don’t need to start from zero everytime.

First understand how you got the formula and then practice well enough to have it on your tips already. One such important part is to able to deal with fractions quickly. This often comes handy in senior years of school, or if you are giving any competitive examination. These are really helpful when you have really odd percentage number to work with in an arithmetic question. For example, you have to find the **83 1/3**% of 18 apples. If you practiced the fraction well, then you will know that **83 1/3**% can be converted into fraction as 5/6, and 5/6 of 18 is **15** and hence the answer.

Here are the rest of fractions that are frequently used:

Fraction | Percentage | Fraction | Percentage | Fraction | Percentage |

1/2 | 50 | 1/12 | 8 1/3 | 1/25 | 4 |

1/3 | 33 1/3 | 1/13 | 7 9/13 | 1/40 | 2 1/2 |

1/4 | 25 | 1/14 | 7 1/7 | 1/80 | 1 1/2 |

1/5 | 20 | 1/15 | 6 2/3 | 1/100 | 1 |

1/6 | 16 2/3 | 1/16 | 6 1/4 | ||

1/7 | 14 2/7 | 1/17 | 5 15/17 | ||

1/8 | 12 1/2 | 1/18 | 5 5/9 | ||

1/9 | 11 1/9 | 1/19 | 5 5/19 | ||

1/10 | 10 | 1/20 | 5 | ||

1/11 | 9 1/11 | 1/24 | 4 1/6 |

8.) **‘x/y + y/x’ Fraction addition**

If the fraction expression is given in the form **x/y + y/x** (say 5/2 + 2/5), then you need to find the square of the difference of **x** and **y** and product of **x** and **y**. Put the value in the expression like this:

x/y + y/x = **{ (x-y)²/xy } + 2**

So, 5/2 + 2/5 = {(2-5)²/(2 x 5)} + 2 = {(-3)²/10} + 2 = 2 (9/10) = **2.9**

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