Three fractions x, y and z are such that x > y > z. When the smallest of them is divided by the greatest, the result is \({\frac{9}{16}}\) , which exceeds y by 0.0625. If x + y + z = \(2{\frac{3}{12}}\) , then what is the value of x + z?
Solving Fraction Equations to Find x + z
The problem provides information about three fractions, x, y, and z, and asks for the value of x + z based on several conditions. We are given that x > y > z, but we will proceed by setting up and solving the equations derived from the other conditions.
Understanding the Given Conditions
We are given the following relationships:
- The smallest fraction (z) divided by the greatest fraction (x) is \({\frac{9}{16}}\). This translates to the equation: $$\frac{z}{x} = \frac{9}{16} \quad (Equation 1)$$
- The result of the division (\({\frac{9}{16}}\)) exceeds y by 0.0625. This means: $$\frac{9}{16} = y + 0.0625 \quad (Equation 2)$$
- The sum of the three fractions is \(2{\frac{3}{12}}\). This means: $$x + y + z = 2{\frac{3}{12}} \quad (Equation 3)$$
Converting Decimal and Mixed Numbers to Fractions
To work with fractions consistently, let's convert the decimal and the mixed number into simple fractions:
- The decimal 0.0625 can be written as: $$0.0625 = \frac{625}{10000} = \frac{125}{2000} = \frac{25}{400} = \frac{1}{16}$$
- The mixed number \(2{\frac{3}{12}}\) can be written as: $$2{\frac{3}{12}} = 2 + \frac{3}{12} = 2 + \frac{1}{4} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4}$$
Now, the equations can be rewritten using only fractions:
- Equation 1: $$\frac{z}{x} = \frac{9}{16}$$
- Equation 2: $$\frac{9}{16} = y + \frac{1}{16}$$
- Equation 3: $$x + y + z = \frac{9}{4}$$
Solving the System of Equations
We can now solve these equations to find the values of x, y, and z.
Step 1: Solve for y using Equation 2.
$$ \frac{9}{16} = y + \frac{1}{16} $$ $$ y = \frac{9}{16} - \frac{1}{16} $$ $$ y = \frac{9-1}{16} = \frac{8}{16} $$ $$ y = \frac{1}{2} $$Step 2: Express z in terms of x using Equation 1.
$$ \frac{z}{x} = \frac{9}{16} $$ $$ z = \frac{9}{16}x $$Step 3: Substitute the values of y and z into Equation 3.
$$ x + y + z = \frac{9}{4} $$ Substitute \(y = \frac{1}{2}\) and \(z = \frac{9}{16}x\): $$ x + \frac{1}{2} + \frac{9}{16}x = \frac{9}{4} $$Step 4: Solve for x.
Combine the terms involving x: $$ x + \frac{9}{16}x = \frac{16}{16}x + \frac{9}{16}x = \frac{16+9}{16}x = \frac{25}{16}x $$ The equation becomes: $$ \frac{25}{16}x + \frac{1}{2} = \frac{9}{4} $$ Subtract \(\frac{1}{2}\) from both sides: $$ \frac{25}{16}x = \frac{9}{4} - \frac{1}{2} $$ Find a common denominator for the right side (\(\frac{1}{2} = \frac{2}{4}\)): $$ \frac{25}{16}x = \frac{9}{4} - \frac{2}{4} $$ $$ \frac{25}{16}x = \frac{9-2}{4} = \frac{7}{4} $$ Multiply both sides by \(\frac{16}{25}\) to isolate x: $$ x = \frac{7}{4} \times \frac{16}{25} $$ $$ x = \frac{7 \times 16}{4 \times 25} = \frac{7 \times 4}{25} $$ $$ x = \frac{28}{25} $$Step 5: Calculate z using the value of x.
Using \(z = \frac{9}{16}x\): $$ z = \frac{9}{16} \times \frac{28}{25} $$ $$ z = \frac{9 \times 28}{16 \times 25} = \frac{9 \times (4 \times 7)}{(4 \times 4) \times 25} = \frac{9 \times 7}{4 \times 25} $$ $$ z = \frac{63}{100} $$Calculating the Value of x + z
Now that we have the values for x and z, we can find their sum:
$$ x + z = \frac{28}{25} + \frac{63}{100} $$ To add these fractions, find a common denominator, which is 100. Convert \(\frac{28}{25}\) to an equivalent fraction with a denominator of 100 by multiplying the numerator and denominator by 4: $$ \frac{28}{25} = \frac{28 \times 4}{25 \times 4} = \frac{112}{100} $$ Now add the fractions: $$ x + z = \frac{112}{100} + \frac{63}{100} $$ $$ x + z = \frac{112 + 63}{100} = \frac{175}{100} $$ Simplify the resulting fraction by dividing the numerator and denominator by their greatest common divisor, which is 25: $$ x + z = \frac{175 \div 25}{100 \div 25} = \frac{7}{4} $$Thus, the value of x + z is \({\frac{7}{4}}\).
Revision Table: Summary of Values
| Fraction | Value | Decimal Value |
|---|---|---|
| x | \(\frac{28}{25}\) | 1.12 |
| y | \(\frac{1}{2}\) | 0.5 |
| z | \(\frac{63}{100}\) | 0.63 |
| x + z | \(\frac{7}{4}\) | 1.75 |
Additional Information on Fraction Problems
Solving problems involving fractions often requires converting between different formats (decimals, mixed numbers, improper fractions) and finding common denominators for addition or subtraction. Setting up equations based on the problem statement is a crucial first step. Once equations are established, standard algebraic techniques can be used to solve for the unknown values. Remember that simplifying fractions at the end is important for presenting the final answer in its simplest form.
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