Concepts of Counting using More or Less Numbers
A place value is given to each digit of a number. Place value is the value indicated by a digit in a number based on its position in the number.
Here’s an illustration of the relationship between the place or position of the digits in a number and their place value.
Thousands | Hundreds | Tens | Ones |
|---|---|---|---|
4 | 9 | 3 | 2 |
Consider the number 4932, and the table shows us how each digit of the number has a specific place value according to where it is placed in a number.
Place value refers to the value of each digit in a number. Understanding place value can help you calculate the value of a number.
Place value is a foundation for the entire numerical system. The value of a number is defined by the position of its digits in this system. Ten of ones comprise one of the next larger unit, tens. Ten of those units make up one of the next larger unit, hundreds.
This pattern applies to both bigger and smaller numbers, or even integers. Add ten to an integer say 3, and it becomes 13, we’re adding 10 ones to 3, pushing the number 3 to represent the number 13 with 3 ones and one tens. If we add 100 to 34, we have 134, which gives us a number with 4 ones, 3 tens and 1 hundreds. In the same way, we can create bigger numbers by adding ten hundreds, which is equal to one thousand.
Consider the number, let’s say 578. This is what the value of each digit would look like. The individual value of each digit is lesser than the previous one, as you move from left to right. Therefore, with respect to their position, the value of each digit is as follows.
From both these representations we can observe that more equals bigger, and less equals smaller.
We know that 3 is greater than 2, when we compare the numbers 3 and 2. To put it another way, 2 is less than 3.
3 is greater than 2, as shown by the “greater than” or “more than” sign, and 2 is less than 3, as indicated by the “less than” sign. We draw these comparisons to understand the value of the numbers provided.
What we will observe next is what happens when we add one to a given number, especially the change in the number, according to its place value.
We can take the same example of 578 once again. If we were to add 1 in the units or ones place, this is how the number would look like.
The number is now 579.
When increasing the tens place by 1, this only means that ten blocks are added to the tens place.
Adding ten blocks that represents adding 1 to the tens place gives us a new number, 588, as ten ones are added to the tens place, altering the number from 578 to 588.
When adding one to the hundreds place, 100 blocks are added to the hundreds place. We saw that the adding 1 gives us 579, and adding 1 to the tens place gives us 588.
Adding 1 to the hundreds place is adding 100 to the hundreds place, which results in 678.
In each of these cases adding 1 to the different numbers that occupy in the specific place values in a number changes the number in different ways. We can also notice that this is addition and adding always increases the value of a number.
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Solved Examples
Example 1:
Johny bought 9 chocolates using the pocket money he saved up. As he came home, his aunt and uncle who came for a visit gave him 11 chocolates. Later that day, Johnny went to the local fair where he took part in a competition and won 888 chocolates. Fill in the table step by step to help Johny keep track of the total number of chocolates he has.
Solution:
Johny bought 9 chocolates initially. We can place it in the column labelled “ones”, as it is a single digit number.
Next, we need to write the number of chocolates Johny’s uncle gave him. Johny’s uncle gave him 11 chocolates . We will write “1” in the one’s place column and the other “1” in the tens place column.
Johny has also won 888 chocolates at the fair. “8” is written on each column of the table.
When adding the total number of chocolates he has so far, we need to add one, nine and eight, which gives us 18, which is 8 ones and one tens.
“1” gets carried over to the tens place, and the total we get by adding the digits in the tens place is ten tens or one hundred.
One gets carried over to the hundreds place. Now, one plus eight is nine, which is nine hundreds.
Hundreds | Tens | Ones |
|---|---|---|
(+1) | (+1) | 9 |
1 | 1 | |
8 | 8 | 8 |
9 | 0 | 8 |
Therefore, the total number of chocolates we get is 908. Johny has a total of 908 chocolates.
Example 2:
568 students were sitting on benches allocated for them across different halls in a building. If each hall holds 100 students and each bench holds 10 students, interpret the number of students in each of the following scenarios.
1. Three benches were added to one of the halls. How many students are totally in the building?
2. Two extra halls were opened for students and they were completely occupied. How many students were there in the building totally?
Solution:
If three benches were added this means that three times 10 students were added to the building,
The total number of students to be added = 10 × 3 = 30 (three tens)
The total number of students = 30 + 568 = 598 (5 hundreds, nine tens, 8 ones)
Table to represent the total number of students when three benches were added:
Hundreds | Tens | Ones |
|---|---|---|
3 | 0 | |
5 | 6 | 8 |
5 | 9 | 8 |
If two halls were occupied with students this means that two times 100 students were added to the building.
The total number of students to be added is = 100 × 2 = 200 (2 hundred)
The total number of students is = 200 + 568 = 768 (7 hundreds, 6 tens, 8 ones)
Table to represent the total number of students when two halls were opened for the students:
Hundreds | Tens | Ones |
|---|---|---|
2 | 0 | 0 |
5 | 6 | 8 |
7 | 6 | 8 |
Example 3:
Janice owns a cupcake shop. She made 120 cupcakes in the morning. By afternoon, 87 cupcakes were sold. Janice made another batch of 60 cupcakes in the afternoon. By evening 90 cupcakes were sold. Figure out the number of cupcakes Janice is left with.
Solution:
Janice made 120 cupcakes in the morning, of which 87 cupcakes were sold.
The number of cupcakes Janice has in the afternoon = 120 – 87 = 33 (3 tens, 3 ones)
Place value chart of the cakes sold and the difference of 33 cakes.
Hundreds | Tens | Ones |
|---|---|---|
1 | 2 | 0 |
8 | 7 | |
3 | 3 |
Janice made another batch of 60 cupcakes in the afternoon.
The number of cupcakes Janice has in the morning = 33 + 60 = 93 (9 tens, 3 ones)
Table showing the number of cupcakes that were there after 60 cupcakes were made.
Hundreds | Tens | Ones |
|---|---|---|
3 | 3 | |
6 | 0 | |
9 | 3 |
Finally, Janice was able to sell another 90 cupcakes.
The number of cupcakes Janice is left with = 93 – 90 = 3
Hundreds | Tens | Ones |
|---|---|---|
9 | 3 | |
9 | 0 | |
3 |
Therefore, Janice has a total of 3 cupcakes left.
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Base ten blocks give a physical understanding of the place value of numbers.
Ten blocks cannot be placed in the ones place as that would shift the number to the left in terms of place value, as well as shift the number ten numbers ahead on the number line. This is the reason why ten blocks cannot be added to the ones place.