Finding Two Numbers from Sum and Difference

This problem asks us to find the ratio of two numbers given their sum and difference. We can solve this by setting up a system of linear equations.

Setting up the Equations for the Two Numbers

Let the two unknown numbers be \(x\)  and  \(y\). According to the problem statement, we have two pieces of information:

1. The sum of the two numbers is 20.
2. Their difference is 2.5.

We can write these statements as equations:

• Equation 1:  \(x + y = 20\)
• Equation 2:  \(x - y = 2.5\)

Solving the System of Equations

We can solve this system using the elimination method. If we add Equation 1 and Equation 2, the  \(y\)  terms will cancel out:

 \((x + y) + (x - y) = 20 + 2.5\) 

 \(x + y + x - y = 22.5\) 

 \(2x = 22.5\) 

Now, we can solve for  \(x\):

 \(x = \frac{22.5}{2}\) 

 \(x = 11.25\) 

Now that we have the value of  \(x\), we can substitute it back into either Equation 1 or Equation 2 to find the value of  \(y\). Let's use Equation 1:

 \(11.25 + y = 20\) 

Subtract 11.25 from both sides to solve for  \(y\):

 \(y = 20 - 11.25\) 

 \(y = 8.75\) 

So, the two numbers are 11.25 and 8.75.

Calculating the Ratio of the Numbers

The question asks for the ratio of these numbers. The ratio of  \(x\)  to  \(y\)  is  \(x : y\). Let's find the ratio of 11.25 to 8.75:

Ratio =  \(11.25 : 8.75\) 

To simplify the ratio, we can first remove the decimals by multiplying both numbers by 100:

Ratio =  \(11.25 \times 100 : 8.75 \times 100\)

Ratio =  \(1125 : 875\)

Now, we can simplify this ratio by finding the greatest common divisor (GCD) or by dividing by common factors. Both numbers end in 5, so they are divisible by 5 and 25. Let's divide by 25:

 \(1125 \div 25 = 45\) 

 \(875 \div 25 = 35\) 

The ratio is now  \(45 : 35\). Both 45 and 35 are divisible by 5:

 \(45 \div 5 = 9\) 

 \(35 \div 5 = 7\) 

The simplified ratio is  \(9 : 7\).

Comparing with Options

Let's compare our calculated ratio with the given options:

• Option 1: 7 : 9
• Option 2: 9 : 7
• Option 3: 3 : 5
• Option 4: 2 : 7

Our calculated ratio of  \(9 : 7\)  matches Option 2.

Step-by-Step Solution Summary

1. Define variables for the two numbers, say  \(x\)  and  \(y\).
2. Formulate equations based on the given sum and difference:  \(x + y = 20\)  and  \(x - y = 2.5\).
3. Solve the system of equations for  \(x\)  and  \(y\). Adding the equations gives  \(2x = 22.5\), so  \(x = 11.25\). Substituting  \(x\)  back gives  \(11.25 + y = 20\), so  \(y = 8.75\).
4. Calculate the ratio  \(x : y\), which is  \(11.25 : 8.75\).
5. Simplify the ratio by multiplying by 100 to remove decimals ( \(1125 : 875\)) and then dividing by common factors (25, then 5) to get  \(9 : 7\).
6. Match the simplified ratio with the given options.

Ratio Calculation and Simplification

The process of simplifying a ratio is similar to simplifying a fraction. We divide both parts of the ratio by the same non-zero number until the parts have no common factors other than 1. In this case, we started with  \(11.25 : 8.75\), which is equivalent to the fraction  \(\frac{11.25}{8.75}\). Multiplying the numerator and denominator by 100 gives  \(\frac{1125}{875}\). Dividing both by 25 gives  \(\frac{45}{35}\). Dividing both by 5 gives  \(\frac{9}{7}\). Thus, the ratio is  \(9 : 7\).

Revision Table: Key Concepts

Additional Information on Word Problems

Word problems often require translating verbal descriptions into mathematical expressions and equations. For problems involving two unknown numbers, information about their sum, difference, product, or quotient is commonly given. Setting up appropriate variables and equations is the first crucial step.

• Sum and Difference: If you know the sum ( \(S\)) and difference ( \(D\)) of two numbers ( \(x, y\)  with  \(x \gt y\)), the numbers can be found directly using the formulas:  \(x = \frac{S + D}{2}\)  and  \(y = \frac{S - D}{2}\).
  In this problem,  \(S = 20\)  and  \(D = 2.5\).
   \(x = \frac{20 + 2.5}{2} = \frac{22.5}{2} = 11.25\) 
   \(y = \frac{20 - 2.5}{2} = \frac{17.5}{2} = 8.75\) 
  This confirms our earlier results obtained by solving the system of equations.
• Ratio and Proportion: Ratios are often used in proportion problems, where two ratios are set equal to each other. Simplifying ratios is a fundamental skill in solving such problems.