If x% of 280 = 15% of 240 + 20% of 310, then the value of x is:
35
Finding the Value of x in a Percentage Equation
Let's solve the given equation step-by-step to find the value of x. The equation is:
\[ x\% \text{ of } 280 = 15\% \text{ of } 240 + 20\% \text{ of } 310 \]Understanding Percentage Calculations
A percentage, like \(y\%\), means \(\frac{y}{100}\). So, \(y\%\) of a number N is calculated as \(\frac{y}{100} \times N\).
Breaking Down the Equation
We need to calculate each term on the right side of the equation first.
Calculating 15% of 240
Using the percentage formula:
\[ 15\% \text{ of } 240 = \frac{15}{100} \times 240 \] \[ = \frac{15}{10} \times 24 \] \[ = \frac{3}{2} \times 24 \] \[ = 3 \times 12 \] \[ = 36 \]So, 15% of 240 is 36.
Calculating 20% of 310
Using the percentage formula:
\[ 20\% \text{ of } 310 = \frac{20}{100} \times 310 \] \[ = \frac{1}{5} \times 310 \] \[ = 62 \]So, 20% of 310 is 62.
Adding the Results
Now, add the calculated values:
\[ 15\% \text{ of } 240 + 20\% \text{ of } 310 = 36 + 62 \] \[ = 98 \]The right side of the original equation equals 98.
Solving for x
The original equation now becomes:
\[ x\% \text{ of } 280 = 98 \]Let's write this in mathematical terms:
\[ \frac{x}{100} \times 280 = 98 \]Now, we solve for x:
\[ x \times \frac{280}{100} = 98 \] \[ x \times \frac{28}{10} = 98 \] \[ x \times \frac{14}{5} = 98 \]Multiply both sides by \(\frac{5}{14}\):
\[ x = 98 \times \frac{5}{14} \]We know that \(98 = 14 \times 7\). Substitute this into the equation:
\[ x = (14 \times 7) \times \frac{5}{14} \]Cancel out the 14:
\[ x = 7 \times 5 \] \[ x = 35 \]Conclusion
The value of x that satisfies the given equation is 35.
| Step | Calculation | Result |
|---|---|---|
| 1 | Calculate 15% of 240 | \(\frac{15}{100} \times 240 = 36\) |
| 2 | Calculate 20% of 310 | \(\frac{20}{100} \times 310 = 62\) |
| 3 | Sum the results | \(36 + 62 = 98\) |
| 4 | Set up equation for x | \(\frac{x}{100} \times 280 = 98\) |
| 5 | Solve for x | \(x = 98 \times \frac{100}{280} = 98 \times \frac{10}{28} = 98 \times \frac{5}{14} = 7 \times 5 = 35\) |
Revision Table: Key Percentage Concepts
| Concept | Formula | Example |
|---|---|---|
| Percentage Definition | \(y\% = \frac{y}{100}\) | \(25\% = \frac{25}{100}\) |
| Finding Percentage of a Number | \(y\%\) of N \( = \frac{y}{100} \times N\) | \(10\%\) of \(50 = \frac{10}{100} \times 50 = 5\) |
| Expressing Fraction/Decimal as Percentage | \(\frac{a}{b} \times 100\%\) or \(0.d \times 100\%\) | \(\frac{1}{4} = \frac{1}{4} \times 100\% = 25\%\) |
Additional Information on Percentage Calculations
Percentages are a fundamental concept in mathematics used widely in various fields like finance, statistics, and everyday calculations. Understanding how to convert between percentages, fractions, and decimals is crucial for solving percentage-based problems.
- To convert a percentage to a decimal, divide by 100. For example, \(35\% = 35/100 = 0.35\).
- To convert a decimal to a percentage, multiply by 100 and add the % symbol. For example, \(0.75 = 0.75 \times 100\% = 75\%\).
- To convert a fraction to a percentage, convert the fraction to a decimal first (by dividing the numerator by the denominator), and then convert the decimal to a percentage. For example, \(\frac{3}{4} = 0.75 = 75\%\).
Solving equations involving percentages often requires setting up the problem correctly by converting percentages to their decimal or fractional equivalents and then using algebraic methods to find the unknown value, as demonstrated in this solution.
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