Finding Two Numbers Based on Sum and Quotient

We are given a problem where we need to find two numbers based on two conditions: their sum and their quotient. Let's denote the two unknown numbers as \(x\) and \(y\).

The problem gives us the following information:

• The sum of the two numbers is 14.
• Their quotient is \(\frac{2}{5}\).

Translating the Problem into Equations

Based on the information provided, we can write two mathematical equations:

1. Sum: The sum of the two numbers \(x\) and \(y\) is 14. \[x + y = 14 \quad \text{(Equation 1)}\]
2. Quotient: The quotient of the two numbers is \(\frac{2}{5}\). This means the ratio of one number to the other is \(\frac{2}{5}\). Without loss of generality, let's assume the ratio of \(x\) to \(y\) is \(\frac{2}{5}\). \[\frac{x}{y} = \frac{2}{5}\] This equation tells us that \(x\) is \(\frac{2}{5}\) times \(y\). We can rearrange this equation to express \(x\) in terms of \(y\) (or vice versa). Multiplying both sides by \(y\), we get: \[x = \frac{2}{5}y \quad \text{(Equation 2)}\] This form is useful for substitution. Note that since the quotient is less than 1, one number must be smaller than the other, and the ratio is likely that of the smaller number to the larger number. Our choice of \(x = \frac{2}{5}y\) reflects this, implying \(x\) is smaller than \(y\).

Solving the System of Equations

We now have a system of two linear equations:

• \(x + y = 14\)
• \(x = \frac{2}{5}y\)

We can solve this system using the substitution method. We will substitute the expression for \(x\) from Equation 2 into Equation 1:

Substitute \(x = \frac{2}{5}y\) into the equation \(x + y = 14\):

\[\left(\frac{2}{5}y\right) + y = 14\]

To combine the terms involving \(y\), we can think of \(y\) as \(\frac{5}{5}y\):

\[\frac{2}{5}y + \frac{5}{5}y = 14\]

Combine the fractions:

\[\frac{2y + 5y}{5} = 14\]

\[\frac{7y}{5} = 14\]

Now, we solve for \(y\). Multiply both sides of the equation by 5:

\[7y = 14 \times 5\]

\[7y = 70\]

Divide both sides by 7:

\[y = \frac{70}{7}\]

\[y = 10\]

Now that we have the value of \(y\), we can find the value of \(x\) by substituting \(y = 10\) back into either Equation 1 or Equation 2. Using Equation 2 (\(x = \frac{2}{5}y\)) is straightforward:

Substitute \(y = 10\) into \(x = \frac{2}{5}y\):

\[x = \frac{2}{5} \times 10\]

\[x = 2 \times \frac{10}{5}\]

\[x = 2 \times 2\]

\[x = 4\]

So, the two numbers are 4 and 10.

Verifying the Numbers

Let's check if the numbers 4 and 10 satisfy the original conditions given in the problem:

• Sum: Is the sum of 4 and 10 equal to 14? \[4 + 10 = 14\] Yes, the sum is 14.
• Quotient: Is the quotient \(\frac{2}{5}\)? The quotient can be \(\frac{4}{10}\) or \(\frac{10}{4}\). \[\frac{4}{10} = \frac{2 \times 2}{2 \times 5} = \frac{2}{5}\] \[\frac{10}{4} = \frac{5 \times 2}{2 \times 2} = \frac{5}{2}\] The problem states "their quotient is \(\frac{2}{5}\)". The ratio of the smaller number (4) to the larger number (10) is indeed \(\frac{2}{5}\).

Both conditions are met by the numbers 4 and 10. Therefore, the two numbers are 10 and 4.

One could also arrive at the answer by testing the given options. For the pair (10, 4):

• Sum: \(10 + 4 = 14\). (Satisfies the sum condition)
• Quotient: Taking the ratio of 4 to 10, we get \(\frac{4}{10} = \frac{2}{5}\). (Satisfies the quotient condition)

This confirms that 10 and 4 are the correct numbers.

Revision Table: Sum and Quotient Problems

Additional Information: Ratios and Parts

The quotient being \(\frac{2}{5}\) means the numbers are in the ratio 2:5. If the numbers are in the ratio \(a:b\), they can be represented as \(ak\) and \(bk\), where \(k\) is a constant of proportionality.

In this case, the ratio is 2:5. So the numbers can be \(2k\) and \(5k\).

Their sum is 14:

Solve for \(k\):

Now find the numbers by substituting the value of \(k\):

First number: \(2k = 2 \times 2 = 4\)

Second number: \(5k = 5 \times 2 = 10\)

The numbers are 4 and 10. This ratio method provides a quick way to solve problems involving sums (or differences) and ratios.