Simplify: \(62\div 5 - \left(\frac{8}{9}\times \frac{18}{7}\right)\times \frac{7}{5}+\frac{5}{4}\times \frac{16}{25}\)
10
Simplifying the Mathematical Expression
To simplify the given expression, we need to follow the order of operations, often remembered by the acronym BODMAS or PEMDAS.
- Brackets / Parentheses
- Orders / Exponents
- Division and Multiplication (from left to right)
- Addition and Subtraction (from left to right)
The expression we need to simplify is:
\(62\div 5 - \left(\frac{8}{9}\times \frac{18}{7}\right)\times \frac{7}{5}+\frac{5}{4}\times \frac{16}{25}\)
Step 1: Simplify within the Parentheses
First, we evaluate the expression inside the parentheses:
\(\left(\frac{8}{9}\times \frac{18}{7}\right)\)
Multiply the fractions:
\(\frac{8}{9}\times \frac{18}{7} = \frac{8 \times 18}{9 \times 7}\)
We can simplify before multiplying by cancelling common factors. \(18\) is \(2 \times 9\):
\(\frac{8 \times (2 \times 9)}{9 \times 7} = \frac{8 \times 2}{7} = \frac{16}{7}\)
Now the expression becomes:
\(62\div 5 - \frac{16}{7}\times \frac{7}{5}+\frac{5}{4}\times \frac{16}{25}\)
Step 2: Perform Division and Multiplication (from left to right)
Next, we perform the division and multiplication operations.
- First term: \(62 \div 5\)
- This can be written as the fraction \(\frac{62}{5}\).
- Second term: \(-\frac{16}{7}\times \frac{7}{5}\)
- Multiply the fractions: \(-\frac{16 \times 7}{7 \times 5}\). The \(7\)s cancel out: \(-\frac{16}{5}\).
- Third term: \(+\frac{5}{4}\times \frac{16}{25}\)
- Multiply the fractions: \(+\frac{5 \times 16}{4 \times 25}\). We can simplify common factors. \(16 = 4 \times 4\) and \(25 = 5 \times 5\): \(+\frac{5 \times (4 \times 4)}{4 \times (5 \times 5)} = +\frac{5 \times 4 \times 4}{4 \times 5 \times 5}\).
- Cancel a \(5\) from numerator and denominator, and a \(4\) from numerator and denominator: \(+\frac{4}{5}\).
The expression is now:
\(\frac{62}{5} - \frac{16}{5} + \frac{4}{5}\)
Step 3: Perform Addition and Subtraction (from left to right)
Since all the fractions have the same denominator (5), we can combine the numerators:
\(\frac{62 - 16 + 4}{5}\)
Calculate the numerator:
\(62 - 16 = 46\)
\(46 + 4 = 50\)
So, the expression simplifies to:
\(\frac{50}{5}\)
Step 4: Final Simplification
Divide 50 by 5:
\(\frac{50}{5} = 10\)
The simplified value of the expression is \(10\).
Revision Table: Order of Operations (BODMAS/PEMDAS)
| Order | Operation | Example |
|---|---|---|
| 1st | Brackets/Parentheses | \((a+b)\) |
| 2nd | Orders/Exponents | \(x^2\), \(\sqrt{x}\) |
| 3rd | Division and Multiplication | \(a \div b\) or \(a \times b\) (Left to Right) |
| 4th | Addition and Subtraction | \(a + b\) or \(a - b\) (Left to Right) |
Additional Information: Working with Fractions
When multiplying fractions, multiply the numerators together and the denominators together. Simplify by cancelling common factors before or after multiplying.
When adding or subtracting fractions, they must have a common denominator. If they do, add or subtract the numerators and keep the denominator the same. If they don't, find a common denominator first.
Important Questions from Fractions
If three-fifths of a number is 54, what is two-ninth of it?
Sunila had \(9\frac{1}{4}\) kg of flour to make bread with. If the recipe says that she needs \(1\frac{1}{8}\) kg to make one loaf of bread, how many loafs can she make? Estimate to the nearest whole number.
The value of \(\frac{{11}}{5} - \left( {\frac{2}{3}of\frac{3}{5} - \frac{1}{5}} \right) + \left( {\frac{6}{5} \div \frac{4}{5}} \right)\) is:
In his career, a tennis player won 5 matches lost 12 matches and had 3 matches as draw. The fraction of the match he lost in his career is:
The value of 8 + \((\frac{1}{2} + \frac{1}{4})\) × 16 is:
