Table of Contents
Place Value
In mathematics, every digit in a number has a place value. The value indicated by a digit in a number is based on its position in the number and is known as its place value.
For example, in 34, the 4 represents 4 ‘ones’ and the 3 represents 3 ‘tens’ or 30.
Associative property
To “associate” with something means to join or link with it. The product of three or more numbers, according to the associative feature of multiplication, remains the same regardless of how the numbers are arranged. If a, b and c are real numbers, then as per the associative property of multiplication;
(a x b) x c = a x (b x c)
Here’s an example of how the product remains the same no matter how the factors are organized.
Take a look at the number line given below. Observe how the pattern of multiplication can be derived from this.
On the number line, we need to move from five to ten. What must be multiplied to five to get the result as ten? Similarly, to get fifteen, what must be multiplied to five?
5 \(\times\) ( ) = 10
5 \(\times\) ( ) = 15
5 \(\times\) ( ) = 20
Five multiplied by two gives us ten. Five multiplied by three gives us fifteen. Five multiplied by four gives us twenty.
5 \(\times\) 2 = 10
5 \(\times\) 3 = 15
5 \(\times\) 4 = 20
Using the number line we can figure out what is twelve times five as well.
Jumping two units from 50 we get 60, which is five times twelve.
Solved Examples
Find the product of the following numbers.
Question 1:
13 \(\times\) 30 = __
Answer:
Let’s find the product using place values,
13 \(\times\) 30 = 13 \(\times\) 3 ‘tens’
= 39 ‘tens’
= 390
So the answer here is 390.
Question 2:
13\(\times\)40 =
Answer:
Let’s find the product using the associative property of multiplication
= 13 \(\times\) 4 \(\times\) 10
We can first multiply four and thirteen which is 52.
= 52 \(\times\) 10
= 520
Question 3:
Use the digits 1,2,3 and 4 in the blanks given below to get the answer shown below.
Answer:
Let’s try arranging the digits 1,2,3 and 4 in different ways to get different numbers and check which arrangement gives us the required product.
13 \(\times\) 24 = 312
12 \(\times\) 34 = 408
32 \(\times\) 14 = 448
We can get various results by doing so but the answer we are looking for is 32.
\(\times\) 14 = 448
Question 4:
1. Use the products of numbers using the digits 1,2,3 and 4 to get a value close to 1000.
Answer:
Let’s try arranging the digits 1,2,3 and 4 in different ways to get different numbers and check which arrangement gives us the required product.
13 \(\times\) 24 = 312
12 \(\times\) 34 = 408
32 \(\times\) 14 = 448
We can get various results by doing so.
We know that 40 \(\times\) 25 = 1000
So let’s try arranging numbers such that they are similar to 40 and 25.
43 \(\times\) 21 = 903
41 \(\times\) 23 = 943
41 \(\times\) 32 = 1312
42 \(\times\) 31 = 1302
Therefore, the value that is closest to 1000 is 943.
Question 5:
Ralph is collecting apples from different houses from his town to make apple custard pies for a charity event. He received 10 apples each from ten houses, 12 apples each from 20 houses and 18 apples each from 12 houses. Find the total number of apples he collected from all the houses.
Answer:
Total number of apples collected from ten houses = 10\(\times\) 10 = 100
Total number of apples collected from twenty houses = 12 \(\times\) 20
Using associative property,
12 \(\times\) 20 = 12 \(\times\) 2 \(\times\) 10
= 24 \(\times\) 10
= 240
Total number of apples collected from twelve houses = 12 \(\times\) 18 = Using associative property,
12 \(\times\) 18 = 10 \(\times\) 8 \(\times\) 10 \(\times\) 2
= 8 \(\times\) 2\(\times\) 10 \(\times\) 10
= 16 \(\times\) 10 \(\times\) 10
= 1600
The total number of apples = 100 + 240 + 1600 = 1940
Therefore the total number of apples is equal to 1940.
The major distinction between place value and face value is that place value assigns a digit’s value based on its position in a number, whereas face value assigns a digit’s real value. A number’s face value is fixed and cannot be changed, however its place value varies depending on the digit’s location.
For example in 25, the place value of 2 is 20 but the face value of 2 is 2.
To “distribute” anything means to divide it or to provide a portion of it.
Multiplying the sum of two or more addends by a number produces the same result as multiplying each addend by the number individually and then adding the products together, according to the distributive property. If a, b and c are whole numbers, the distributive property is as follows.
a x (b + c) = a x b + a x c
Here is an example that shows the distributive property.
