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5th Grade Patterns in Whole Numbers
In 5th grade patterns in whole numbers we will learn patterns in multiples of 3, patterns in the Associative property of whole numbers, patterns in the Distributive property of whole numbers, patterns in addition and subtraction of whole numbers, patterns in multiplication of whole numbers and different types of solve examples on patterns of whole numbers.
Patterns In Multiples of 3:
Consider some numbers which are multiples of 3:
12, 18, 27, 87, 99, 102, 111, 306, 615, 2463
Now, add the digits of these numbers. If the sum of the digits is a two-digit number, then further add its digits. What do you observe?
The sum of the digits is 3, 6 or 9. Thus, we can say that the sum of the digits of any multiples of 3 is also a multiple of 3.
This property of whole numbers can be used to check whether a given number is a multiple of 3 or not without actual division.
Patterns in the Associative Property:
Associative property of addition and multiplication of whole numbers can be used to find the sum of numbers easily.
Consider the three numbers 89, 346 and 11.
To find the sum of these numbers, we can proceed as follows:
89 + 346 + 11 = (89 + 11) + 346
= 100 + 346
= 446
Similarly, 783 + 2945 + 217 = (783 + 217) + 2945
= 1000 + 2945
= 3945
To find the product of 125, 378 and 8, proceed as follows:
125 × 378 × 8 = (125 × 8) × 378
= 1000 × 378
= 378000
Similarly, 50 × 974 × 2 = (50 × 2) × 974
= 100 × 974
= 97400
Patterns in the Distributive Property:
Arithmetic calculation can be done easily using the distributive property of multiplication over addition.
Let us consider the following simplification:
897 × 13 + 87 × 897
Then, 897 × 13 + 87 × 897
= 897(13 + 87)
= 897 × 100 = 89700
Similarly, 723 × 956 + 44 × 723
= 723(956 + 44)
= 723 × 1000
= 723000
Patterns in Addition and Subtraction:
We know that 10 - 2 = 9 - 1 = 8 and 93 - 4 = 92 - 3 = 89
This pattern can be used to subtract numbers easily.
For example:
1000 - 786 = 999 - 785 = 214
7900 - 2796 = 7899 - 2795 = 5104
Similarly, 2798 + 998 = 2798 + 1000 - 2
= 3796
and 1234 - 99 = 1234 - 100 + 1
= 1135
Patterns in Multiplication:
Observe the following to explore the patterns in the multiplication:
(i) Product of a Number by another Number Ending with 5:
For Example:
482 × 5 = 482 × \(\frac{10}{2}\) = 2410
960 × 25 = 960 × \(\frac{100}{4}\) = 2400
72 × 15 = 72 × \(\frac{30}{2}\) = 36 × 30 = 1080
(ii) Product of a 2-digit or 3-digit Number Ending with 5 by Itself:
|
15 × 15 225 ↓ 1 × 2 |
25 × 25 625 ↓ 2 × 3 |
35 × 35 1225 ↓ 3 × 4 |
105 × 105 11025 ↓ 10 × 11 |
(ii) Simplification Involving Multiplication, Addition and Subtraction
(2 × 2) - (1 × 1) = 4 - 1 = 3 = 2 + 1
(3 × 3) - (2 × 2) = 9 - 4 = 5 = 3 + 2
(4 × 4) - (3 × 3) = 16 - 9 = 7 = 4 + 3
From this pattern, we get
(101 × 101) - (100 × 100) = 101 + 100 = 201 and
(999 × 999) - (998 × 998) = 999 + 998 = 1000 - 1 + 1000 - 2 = 2000 - 3 = 1997
Solved Examples on 5th Grade Patterns in Whole Numbers:
1. Check whether the following numbers are multiples of 3.
(i) 714
(ii) 2056
Solution:
(i) Sum of the digits of 714 = 7 + 1 + 4 = 12,
then sum of the digits of 12 = 1 + 2 = 3
So, we can say that 714 is a multiple of 3.
(ii) Sum of the digits of 2056 = 2 + 0 + 5 + 6 = 13,
then sum of the digits of 13 = 1 + 3 = 4.
Since 4 is a not a multiple of 3, therefore 2056 is not a multiple of 3.
2. Simplify the following problems using appropriate properties:
(i) 1895 + 3491 + 105
(ii) 4 × 479 × 25
(iii) 67891 × 67 + 67891 × 33
(iv) 9756 × 116 - 16 × 9756
(v) 1723 × 102
(vi) 1489 × 99
Solution:
(i) 1895 + 3491 + 105 = (1895 + 105) + 3491
= 2000 + 3491
= 5491
(ii) 4 × 479 × 25 = (4 × 25) × 479
= 100 × 479
= 47900
(iii) 67891 × 67 + 67891 × 33 = 67891 × (67 + 33)
= 67891 × 100
= 6789100
(iv) 9756 × 116 - 16 × 9756 = 9756 × (116 - 16)
= 9756 × 100
= 975600
(v) 1723 × 102 = 1723 (100 + 2)
= 1723 × 100 + 1723 × 2
= 172300 + 3446
= 175746
(vi) 1489 × 99 = 1489 × (100 - 1)
= 148900 - 1489
= 147411
3. Find the difference using patterns:
(i) 8000 - 2475
(ii) 10000 - 7658
Solution:
(i) 8000 - 2475 = 7999 - 2474
= 5525
(ii) 10000 - 7658 = 9999 - 7657
= 2342
4. Evaluate the following using suitable patterns:
(1) 1979 + 99
(ii) 2013 + 199
(iii) 848 × 25
(iv) 128 × 125
Solution:
(i) 1979 + 99 = 1979 + 100 - 1
= 2079 - 1
= 2078
(ii) 2013 + 199 = 2013 + 200 - 1
= 2213 - 1
= 2212
(iii) 848 × 25 = 848 × \(\frac{100}{4}\)
= 212 × 100
= 21200
(iv) 128 × 125 = 128 × \(\frac{1000}{8}\)
= 16000
5. Evaluate the following using patterns:
(i) 95 × 95
(ii) 205 × 205
(iii) (89 × 89) + (88 × 8)
(iv) (2000 × 2000) - (1999 × 1999)
Solution:
|
(i) |
95 × 95 9025 ↓ 9 × 10 |
(ii) |
205 × 205 42025 ↓ 20 × 21 |
(ii) (89 × 89) + (88 × 88) = 89 + 88 = 177
(iv) (2000 × 2000) - (1999 × 1999) = 2000 + 1999 = 3999
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