The sum of two numbers is 66 and their difference is 22. What is the ratio of the two numbers?
2 ∶ 1
Finding the Ratio of Two Numbers Given Sum and Difference
The question asks us to find the ratio of two numbers when their sum and difference are given. This is a common type of algebra problem that involves setting up and solving a system of linear equations.
Setting Up the Equations
Let the two numbers be represented by variables, say $x$ and $y$.
We are given two pieces of information:
- The sum of the two numbers is 66.
- The difference of the two numbers is 22.
We can translate these statements into mathematical equations:
Equation 1: The sum is 66
\( x + y = 66 \)
Equation 2: The difference is 22
\( x - y = 22 \)
Solving the System of Equations
We now have a system of two linear equations with two variables:
\( x + y = 66 \quad (1) \)
\( x - y = 22 \quad (2) \)
There are several ways to solve this system, such as substitution or elimination. The elimination method is straightforward here because the $y$ terms have opposite signs in the two equations.
Add Equation (1) and Equation (2):
\( (x + y) + (x - y) = 66 + 22 \)
\( x + y + x - y = 88 \)
\( 2x = 88 \)
Now, solve for $x$ by dividing both sides by 2:
\( x = \frac{88}{2} \)
\( x = 44 \)
Now that we have the value of $x$, substitute it back into either Equation (1) or Equation (2) to find the value of $y$. Let's use Equation (1):
\( x + y = 66 \)
\( 44 + y = 66 \)
Subtract 44 from both sides to solve for $y$:
\( y = 66 - 44 \)
\( y = 22 \)
So, the two numbers are 44 and 22.
Finding the Ratio of the Two Numbers
The question asks for the ratio of the two numbers. The ratio of $x$ to $y$ is written as $x : y$ or $\frac{x}{y}$.
The ratio of the two numbers (44 and 22) is:
\( \text{Ratio} = 44 : 22 \)
To simplify the ratio, find the greatest common divisor (GCD) of 44 and 22. The GCD of 44 and 22 is 22.
Divide both parts of the ratio by 22:
\( \frac{44}{22} : \frac{22}{22} \)
\( 2 : 1 \)
The ratio of the two numbers is 2 : 1.
Verification
Let's check if these two numbers (44 and 22) satisfy the original conditions:
- Sum: \( 44 + 22 = 66 \). This is correct.
- Difference: \( 44 - 22 = 22 \). This is correct.
The ratio is indeed 2 : 1.
| Step | Description | Result |
|---|---|---|
| 1 | Define variables for the two numbers | \(x, y\) |
| 2 | Set up equations from sum and difference | \(x+y=66\), \(x-y=22\) |
| 3 | Solve the system for \(x\) | \(x=44\) |
| 4 | Solve the system for \(y\) | \(y=22\) |
| 5 | Form the ratio \(x:y\) | \(44:22\) |
| 6 | Simplify the ratio | \(2:1\) |
Revision Table: Sum and Difference Problems
| Concept | Key Idea | Application in this problem |
|---|---|---|
| Representing Unknowns | Use variables (e.g., \(x\), \(y\)) for unknown numbers. | Let the numbers be \(x\) and \(y\). |
| Translating Words to Equations | "Sum" means addition (+), "difference" means subtraction (-). | \(x+y = \text{sum}\), \(x-y = \text{difference}\). |
| Solving System of Equations | Methods like substitution or elimination help find variable values. | Used elimination by adding equations to find \(x\), then substituted to find \(y\). |
| Ratio Calculation | A ratio \(a:b\) is the division \(\frac{a}{b}\) in simplest form. | Calculated \(44:22\) and simplified by dividing by the GCD (22). |
Additional Information: Ratio Concepts
- A ratio is a comparison of two quantities.
- Ratios can be written in several ways: \(a:b\), $a$ to $b$, or $\frac{a}{b}$ (as a fraction).
- Simplifying a ratio means dividing both parts by their greatest common divisor (GCD) to get the smallest possible whole number ratio.
- When a problem gives the sum and difference of two numbers, you can quickly find the numbers:
- Larger number = (Sum + Difference) / 2
- Smaller number = (Sum - Difference) / 2
- Using the quick method for this problem:
- Larger number = \((66 + 22) / 2 = 88 / 2 = 44\)
- Smaller number = \((66 - 22) / 2 = 44 / 2 = 22\)
- This confirms our numbers are 44 and 22, and their ratio is \(44:22 = 2:1\).
Important Questions from Linear Equation in 2 Variable
If 2 x + 3 y = 17;
2 x+2 - 3 y+1 = 5
then the values of x and y are:
The solution of pair of linear equations \(\dfrac{1}{2}x+\dfrac{2}{3}y=-1,x-\dfrac{1}{3}y=3\) by the elimination method, is:
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