What is the value of \(\frac{156}{13} × \frac{147}{24}× \frac{8}{49}\)of \(\frac{24}{144}\)?
2
Simplifying the Fraction Expression to Find the Value
The question asks for the value of a mathematical expression involving fractions and the word "of". In mathematics, the word "of" signifies multiplication. So, the expression can be rewritten as a product of four fractions.
The given expression is:
\(\frac{156}{13} \times \frac{147}{24} \times \frac{8}{49}\) of \(\frac{24}{144}\)
Replacing "of" with multiplication, we get:
\(\frac{156}{13} \times \frac{147}{24} \times \frac{8}{49} \times \frac{24}{144}\)
Now, we can simplify this product by cancelling common factors between the numerators and denominators. It's often helpful to simplify individual fractions first, or cancel terms across the multiplication.
Step-by-Step Calculation
Let's simplify the expression:
\(\frac{156}{13} \times \frac{147}{24} \times \frac{8}{49} \times \frac{24}{144}\)
- Simplify \(\frac{156}{13}\): \(156 \div 13 = 12\). The expression becomes: \(12 \times \frac{147}{24} \times \frac{8}{49} \times \frac{24}{144}\).
- Notice that we have \(24\) in the denominator of the second term and \(24\) in the numerator of the fourth term. These can cancel out: \(12 \times \frac{147}{\cancel{24}} \times \frac{8}{49} \times \frac{\cancel{24}}{144}\). The expression is now: \(12 \times \frac{147}{49} \times \frac{8}{144}\).
- Simplify \(\frac{147}{49}\): \(147 \div 49 = 3\) (since \(3 \times 49 = 147\)). The expression becomes: \(12 \times 3 \times \frac{8}{144}\).
- Simplify \(\frac{8}{144}\): Both 8 and 144 are divisible by 8. \(8 \div 8 = 1\), \(144 \div 8 = 18\). The expression becomes: \(12 \times 3 \times \frac{1}{18}\).
- Now, multiply the remaining terms: \(12 \times 3 \times \frac{1}{18}\). This is equal to \(\frac{12 \times 3}{18}\).
- Calculate the product in the numerator: \(12 \times 3 = 36\). The expression is \(\frac{36}{18}\).
- Simplify the final fraction: \(36 \div 18 = 2\).
So, the value of the expression is 2.
Summary of Simplification Steps
| Step | Expression | Simplification | Result |
|---|---|---|---|
| 1 | \(\frac{156}{13} \times \frac{147}{24} \times \frac{8}{49} \times \frac{24}{144}\) | \(\frac{156}{13} = 12\) | \(12 \times \frac{147}{24} \times \frac{8}{49} \times \frac{24}{144}\) |
| 2 | \(12 \times \frac{147}{24} \times \frac{8}{49} \times \frac{24}{144}\) | Cancel \(\cancel{24}\) | \(12 \times \frac{147}{49} \times \frac{8}{144}\) |
| 3 | \(12 \times \frac{147}{49} \times \frac{8}{144}\) | \(\frac{147}{49} = 3\) | \(12 \times 3 \times \frac{8}{144}\) |
| 4 | \(12 \times 3 \times \frac{8}{144}\) | \(\frac{8}{144} = \frac{1}{18}\) | \(12 \times 3 \times \frac{1}{18}\) |
| 5 | \(12 \times 3 \times \frac{1}{18}\) | Multiply numerators | \(\frac{36}{18}\) |
| 6 | \(\frac{36}{18}\) | Simplify fraction | \(2\) |
The final value of the given expression is 2.
Revision Table: Key Math Concepts
| Concept | Description | Example |
|---|---|---|
| Fraction | Represents a part of a whole; written as numerator/denominator. | \(\frac{1}{2}\), \(\frac{3}{4}\) |
| Multiplication of Fractions | Multiply numerators together and denominators together. \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\) | \(\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}\) |
| Word "of" | Means multiplication in mathematical expressions. | "half of 10" means \(\frac{1}{2} \times 10\) |
| Simplification (Cancelling) | Dividing a numerator and a denominator by a common factor before multiplying to make calculation easier. | \(\frac{\cancel{2}}{3} \times \frac{1}{\cancel{2}} = \frac{1}{3}\) |
Additional Information: Simplifying Complex Expressions
Simplifying complex mathematical expressions involving fractions, decimals, or mixed operations requires following the order of operations (BODMAS/PEMDAS) and applying the rules for each operation correctly. For expressions with fractions, cancelling common factors before multiplying is a very useful technique that reduces the size of numbers and makes the calculation less prone to errors. Always look for opportunities to simplify fractions or cancel terms across multiplication signs.
Important Questions from Fractions
Which fraction among the following is the least ?
\(\frac{5}{11}, \frac{7}{12}, \frac{8}{13}, \frac{9}{17}\)
Find the value of the following expression:
\(\frac{{3 \div 1 \times 2 + 5 - 2}}{{3 \times 3 - 2}}\)
Simplify the expression 441 ÷ \(\left[270 \div \frac{3}{7}+\left(17\div \frac{1}{3}\right)-\left(8\frac{1}{2}-\frac{5}{2}\right)\right]\)
If the sum of two positive numbers is 65 and the square root of their product is 26, then the sum of their reciprocals is:
The value of \(9 \div [\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{6}\div(\frac{3}{4}-\frac{1}{3})\;of\;\frac{2}{9}]\) is:
