Two numbers are in the ratio of 9 : 7. If the larger number is 56 more than one-seventh of the smaller, then what is the sum of the two numbers?
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Understanding the Ratio Problem
The question gives us information about two numbers. First, their relationship is described by a ratio. The ratio of the two numbers is given as 9 : 7. This means that if we represent the numbers using a common factor, say \(x\), the larger number can be written as \(9x\) and the smaller number can be written as \(7x\).
Second, a specific condition is given about the larger number in relation to the smaller number. The larger number is 56 more than one-seventh of the smaller number. We can translate this condition into an algebraic equation to find the value of \(x\).
Our goal is to find the sum of the two numbers, which will be \(9x + 7x = 16x\).
Setting Up the Equation from the Number Relationship
Let the larger number be \(L\) and the smaller number be \(S\).
- From the ratio, we have \(\frac{L}{S} = \frac{9}{7}\). This implies \(L = 9x\) and \(S = 7x\) for some common factor \(x\).
- The condition states: The larger number is 56 more than one-seventh of the smaller number.
Translating the condition into an equation:
Larger number = (One-seventh of the smaller number) + 56
Substituting our expressions for \(L\) and \(S\) in terms of \(x\):
\[ 9x = \frac{1}{7} \times (7x) + 56 \]Solving for the Unknown 'x'
Now we need to solve the equation \(9x = \frac{1}{7}(7x) + 56\) to find the value of \(x\).
\[ 9x = \frac{7x}{7} + 56 \] \[ 9x = x + 56 \]To isolate the term with \(x\), we subtract \(x\) from both sides of the equation:
\[ 9x - x = 56 \] \[ 8x = 56 \]Now, to find \(x\), we divide both sides by 8:
\[ x = \frac{56}{8} \] \[ x = 7 \]So, the common factor \(x\) is 7.
Finding the Two Numbers
Using the value \(x = 7\), we can find the actual values of the two numbers:
- Larger number = \(9x = 9 \times 7 = 63\)
- Smaller number = \(7x = 7 \times 7 = 49\)
Let's quickly check if these numbers satisfy the given condition: Is the larger number (63) equal to 56 more than one-seventh of the smaller number (49)?
One-seventh of the smaller number = \(\frac{1}{7} \times 49 = 7\)
56 more than one-seventh of the smaller number = \(7 + 56 = 63\)
Yes, 63 = 63. The condition is satisfied.
Calculating the Sum of the Two Numbers
The question asks for the sum of the two numbers. The numbers are 63 and 49.
Sum = Larger number + Smaller number
Sum = 63 + 49
Sum = 112
Alternatively, the sum is \(16x\). Using \(x=7\), the sum is \(16 \times 7 = 112\).
Summary of Steps to Solve Ratio Problems
Here is a breakdown of the approach used to solve this problem:
- Represent the numbers using the given ratio and a common variable \(x\).
- Translate the given condition relating the numbers into an algebraic equation involving \(x\).
- Solve the equation for \(x\).
- Substitute the value of \(x\) back into the expressions for the numbers to find their actual values.
- Perform the final calculation requested (in this case, find the sum).
| Concept | Description | Application in Problem |
|---|---|---|
| Ratio | A comparison of two quantities. | Numbers are in ratio 9:7, represented as \(9x\) and \(7x\). |
| Algebraic Equation | A mathematical statement that two expressions are equal, often involving variables. | \(9x = \frac{1}{7}(7x) + 56\). |
| Solving Equation | Finding the value(s) of the variable(s) that satisfy the equation. | Solving \(8x = 56\) to find \(x=7\). |
| Sum of Numbers | The result of adding the two numbers together. | \(63 + 49 = 112\). |
Revision Table: Key Concepts in Ratio and Number Problems
| Term | Definition | Example |
|---|---|---|
| Ratio | A way to compare two or more quantities. Written as a:b or a/b. | If apples to bananas are 3:2, there are 3 apples for every 2 bananas. |
| Proportion | An equation stating that two ratios are equal. | \(a/b = c/d\). Used to find an unknown quantity in a ratio. |
| Algebraic Expression | A combination of variables, numbers, and arithmetic operations. | \(9x\), \(7x\), \(\frac{1}{7}y + 56\). |
| Variable | A symbol (like \(x\) or \(y\)) that represents a quantity that can change. | In \(9x\) and \(7x\), \(x\) is the variable representing the common factor. |
Additional Information: Solving Number Relation Problems
Problems involving relationships between numbers, often described using phrases like "more than," "less than," "times," "one-third of," etc., can usually be solved by setting up an algebraic equation. The key steps are:
- Identify the unknown quantity or quantities.
- Assign variables to represent these unknowns.
- Translate the given verbal statements into mathematical expressions and equations.
- Solve the equation(s) using algebraic techniques.
- Check your answer by substituting the values back into the original problem statement.
When dealing with ratios, representing the numbers using a common multiple (\(ax\), \(bx\), etc.) is a standard and effective method to simplify the problem and translate it into a single-variable equation.
Important Questions from Simple Ratios
If x : y = 5 : 2, then the value of (8x + 9y) : (8x + 2y) is:
If the value of \(\frac{{p - q}}{q} = \frac{4}{3}\) .find p : q
The ratio of monthly income and expenditure of a person is 57 : 43. If he saves Rs. 42,000 per annum, What is his monthly income?
The ratio of two numbers A and B is 5: 8. If 5 is added to each of A and B, then the ratio becomes 2 : 3. The difference between A and B is:
In population of a city the ratio of men and women is 12 : 11. If total population of that city is 4,60,000, then the population of women is:
