How many one-thirds are there in 72?
216
Understanding the Question: How Many One-Thirds are in 72?
The question asks how many times the fraction "one-third" (\(\frac{1}{3}\)) fits into the whole number 72. This type of problem requires us to perform division. We need to divide the total quantity (72) by the size of the part we are counting (\(\frac{1}{3}\)).
Step-by-Step Calculation to Find the Number of One-Thirds
To find out how many one-thirds are in 72, we set up the division problem:
Number of one-thirds = \(72 \div \frac{1}{3}\)
Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of \(\frac{1}{3}\) is \(\frac{3}{1}\) or simply 3.
So, the calculation becomes:
Number of one-thirds = \(72 \times 3\)
Now, we perform the multiplication:
- Multiply 72 by 3.
- \(72 \times 3 = (70 + 2) \times 3 = (70 \times 3) + (2 \times 3)\)
- \(70 \times 3 = 210\)
- \(2 \times 3 = 6\)
- \(210 + 6 = 216\)
Therefore, there are 216 one-thirds in 72.
Checking the Options
Let's compare our calculated result with the given options:
| Option | Value | Matches Calculation? |
|---|---|---|
| 1 | 216 | Yes |
| 2 | 24 | No |
| 3 | 144 | No |
| 4 | 288 | No |
Our calculated value, 216, matches Option 1.
Conclusion on Number of One-Thirds in 72
Based on the division of 72 by one-third, we find that there are 216 one-thirds in 72. This aligns with Option 1.
Revision Table: Key Concepts for Finding One-Thirds
| Concept | Explanation | How it applies here |
|---|---|---|
| Fraction | A part of a whole, like \(\frac{1}{3}\). | The question asks about counting parts (\(\frac{1}{3}\)). |
| Division | Splitting a number into equal parts, or finding how many times one number fits into another. | We divide the whole (72) by the size of the part (\(\frac{1}{3}\)). |
| Reciprocal | Flipping a fraction (numerator becomes denominator and vice versa). Reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\). | To divide by a fraction, multiply by its reciprocal. Reciprocal of \(\frac{1}{3}\) is 3. |
| Multiplication | Repeated addition, or scaling a number. | We multiply 72 by 3 after changing the division problem. |
Additional Information on Fractions and Division
When you ask "how many X are in Y?", you are essentially performing the operation \(Y \div X\).
For example:
- How many 2s are in 10? \(10 \div 2 = 5\). There are five 2s in 10 (\(2+2+2+2+2=10\)).
- How many apples are in 1 dozen? A dozen is 12 apples. \(12 \div 1 = 12\). There are twelve 1-apple units in 12 apples.
- How many quarters are in 1 dollar? A quarter is $0.25. \(1 \div 0.25 = 4\). There are four quarters in a dollar. This is equivalent to \(1 \div \frac{1}{4} = 1 \times 4 = 4\).
Similarly, asking how many one-thirds are in 72 means calculating \(72 \div \frac{1}{3}\). This confirms the approach used in the solution.
Important Questions from Integers
Find the number of integers between $1$ and $150$ (inclusive) having $7$ as one of the digits but which are not divisible by $7$.
Integers are listed from 700 to 1000. In how many integers is the sum of the digits 10 ?
Using 2, 2, 3, 3, 3 as digits, how many distinct numbers greater than 30000 can be formed ?
Consider the following statements :
1. The sum of 5 consecutive integers can be 100.
2 The product of three consecutive natural numbers can be equal to their sum.
Which of the above statements is/are correct ?
The difference between a 2-digit number and the number obtained by interchanging the positions of the digits is 54.
Consider the following statements:
1. The sum of the two digits of the number can be determined only if the product of the two digits is known.
2. The difference between the two digits of the number can be determined.
Which of the above statements is/are correct?
