If the first number is 20% less than the third number and the ratio of the second to the third number is 7 ∶ 10, then the average of the first and the third number is how much per cent more than the second number?
Solving Number Percentage and Ratio Problems
This problem involves understanding percentages and ratios relating three different numbers and then calculating an average and a percentage difference. Let's break it down step by step to find the solution.
Defining the Numbers and Relationships
Let the three numbers be represented as follows:
- First Number: \(N_1\)
- Second Number: \(N_2\)
- Third Number: \(N_3\)
According to the question, we have two main relationships:
- The first number (\(N_1\)) is 20% less than the third number (\(N_3\)).
- The ratio of the second number (\(N_2\)) to the third number (\(N_3\)) is 7 ∶ 10.
Translating Relationships into Mathematical Terms
We can write these relationships as equations:
- \(N_1\) is 20% less than \(N_3\): \(N_1 = N_3 - 20\% \text{ of } N_3\) \(N_1 = N_3 - 0.20 \times N_3\) \(N_1 = N_3 (1 - 0.20)\) \(N_1 = 0.80 \times N_3\)
- Ratio of \(N_2\) to \(N_3\) is 7 ∶ 10: \(\frac{N_2}{N_3} = \frac{7}{10}\) \(N_2 = \frac{7}{10} \times N_3\) \(N_2 = 0.7 \times N_3\)
Using an Example Value to Calculate the Numbers
To make calculations easier, we can assume a value for \(N_3\). A good choice is a number divisible by 10 (from the ratio) or 100 (for percentages). Let's assume \(N_3 = 100\).
- If \(N_3 = 100\), then \(N_1 = 0.80 \times 100 = 80\).
- If \(N_3 = 100\), then \(N_2 = 0.7 \times 100 = 70\).
So, based on \(N_3 = 100\), the three numbers are 80, 70, and 100.
Calculating the Average of the First and Third Numbers
The average of the first number (\(N_1\)) and the third number (\(N_3\)) is:
Average \( = \frac{N_1 + N_3}{2}\)
Using our example values:
Average \( = \frac{80 + 100}{2} = \frac{180}{2} = 90\)
Comparing the Average to the Second Number
We need to find out how much per cent the average (90) is more than the second number (70).
Difference \( = \text{Average} - N_2 = 90 - 70 = 20\)
The difference is 20. To find the percentage increase compared to the second number, we use the formula:
Percentage Increase \( = \left( \frac{\text{Difference}}{N_2} \right) \times 100\%\)
Percentage Increase \( = \left( \frac{20}{70} \right) \times 100\%\)
Percentage Increase \( = \frac{2}{7} \times 100\%\)
Percentage Increase \( = \frac{200}{7}\%\)
Converting the Percentage to a Mixed Number
Now, let's convert the improper fraction \(\frac{200}{7}\) into a mixed number:
\(\frac{200}{7} = 200 \div 7\)
Dividing 200 by 7:
\(200 = 7 \times 28 + 4\)
So, \(\frac{200}{7}\) can be written as \(28\) with a remainder of \(4\). The mixed number is \(28 \frac{4}{7}\).
The average of the first and third number is \(28 \frac{4}{7}\) per cent more than the second number.
| Number | Value (assuming \(N_3=100\)) | How it's calculated |
|---|---|---|
| Third Number (\(N_3\)) | 100 | Assumed value |
| First Number (\(N_1\)) | 80 | 20% less than \(N_3\) (\(0.80 \times 100\)) |
| Second Number (\(N_2\)) | 70 | 7/10 of \(N_3\) (\(0.7 \times 100\)) |
| Average of \(N_1\) and \(N_3\) | 90 | \(\frac{80+100}{2}\) |
| Difference (Average - \(N_2\)) | 20 | \(90 - 70\) |
| Percentage More | \(28 \frac{4}{7}\%\) | \(\left(\frac{20}{70}\right) \times 100\%\) |
Revision Table: Key Concepts
| Concept | Explanation | Formula/Method |
|---|---|---|
| Percentage Less Than | If A is X% less than B, then \(A = B \times (1 - \frac{X}{100})\). | \(A = B - B \times \frac{X}{100}\) |
| Ratio | A ratio A:B means the fraction A/B. If \(A:B = x:y\), then \(\frac{A}{B} = \frac{x}{y}\). | Direct proportion |
| Average | The sum of numbers divided by the count of numbers. | Average \( = \frac{\text{Sum of numbers}}{\text{Count of numbers}}\) |
| Percentage Change | Describes how much a value has changed relative to an original value. | Percentage Change \( = \left( \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \right) \times 100\%\) (Use difference if asking 'how much more/less') |
Additional Information: Percentage and Ratio Applications
Percentage and ratio concepts are fundamental in many areas of mathematics and real life, including finance, economics, and statistics. Understanding how to convert between fractions, decimals, and percentages is crucial for solving these types of problems efficiently.
- Percentages represent a part out of 100. For example, 20% is equivalent to \(\frac{20}{100}\) or 0.20.
- Ratios compare two or more quantities. A ratio of 7:10 means for every 7 units of the first quantity, there are 10 units of the second.
- When solving problems involving percentages and ratios, it is often helpful to assume a convenient value for one variable (like 100 or a common multiple) to simplify calculations.
Important Questions from Percentage
Radha saves 25% of her income. If her expenditure increases by 20% and her income increases by 29%, then her savings increase by;
The income of A is 45% more than the income of B and the income of C is 60% less than the sum of the incomes of A and B. The income of D is 20% more than that of C. If the difference between the incomes of B and D is Rs. 13200, then the income (in Rs.) of C is:
The price of cooking oil increased by 25%. Find by how much percentage a family must reduce its consumption in order to maintain the same budget.
The population of a city increased by 30% in the first year and decreased by 15% in the next year. If the present population is 11,050 then population 2 years ago was:
The income of A is 30% less than the income of B and the income of B is 137.5% more than that of C. If the income of A is Rs. 28500 less than that of B, then the income (in Rs.) of C is:
