If the points P and Q represent the real numbers 0.83 ̅ and 0.62 ̅ on the number line, then the distance between P and Q is
19/90
Calculating Distance on the Number Line
The problem asks for the distance between two points, P and Q, on a number line. These points represent the real numbers \(0.83\bar{}\) and \(0.62\bar{}\) respectively. To find the distance between two points on a number line, we calculate the absolute difference between the real numbers they represent.
The first step is to convert the given repeating decimals into fractions.
Converting Repeating Decimals to Fractions
Let's convert the number represented by point P, which is \(0.83\bar{}\), into a fraction.
- Let \(x = 0.83\bar{}\). This means \(x = 0.8333...\).
- Multiply by 10 to move the non-repeating part (\(8\)) before the decimal: \(10x = 8.333...\) (Equation 1)
- Multiply by 100 to move one cycle of the repeating part (\(3\)) before the decimal: \(100x = 83.333...\) (Equation 2)
- Subtract Equation 1 from Equation 2: \(100x - 10x = 83.333... - 8.333...\) \(90x = 75\)
- Solve for \(x\): \(x = \frac{75}{90}\)
- Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 15: \(x = \frac{75 \div 15}{90 \div 15} = \frac{5}{6}\)
So, point P represents the fraction \(\frac{5}{6}\).
Next, let's convert the number represented by point Q, which is \(0.62\bar{}\), into a fraction.
- Let \(y = 0.62\bar{}\). This means \(y = 0.6222...\).
- Multiply by 10 to move the non-repeating part (\(6\)) before the decimal: \(10y = 6.222...\) (Equation 3)
- Multiply by 100 to move one cycle of the repeating part (\(2\)) before the decimal: \(100y = 62.222...\) (Equation 4)
- Subtract Equation 3 from Equation 4: \(100y - 10y = 62.222... - 6.222...\) \(90y = 56\)
- Solve for \(y\): \(y = \frac{56}{90}\)
- Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 2: \(y = \frac{56 \div 2}{90 \div 2} = \frac{28}{45}\)
So, point Q represents the fraction \(\frac{28}{45}\).
Calculating the Distance Between P and Q
The distance between two points P and Q on a number line is given by the absolute difference between the numbers they represent. Let the number for P be \(x_P = \frac{5}{6}\) and the number for Q be \(x_Q = \frac{28}{45}\).
Distance = \(|x_P - x_Q|\) or \(|x_Q - x_P|\)
Let's calculate the difference \(x_P - x_Q\):
\[ \text{Difference} = \frac{5}{6} - \frac{28}{45} \]To subtract fractions, we need a common denominator. The least common multiple (LCM) of 6 and 45 is 90.
Convert the fractions to have a denominator of 90:
\[ \frac{5}{6} = \frac{5 \times 15}{6 \times 15} = \frac{75}{90} \] \[ \frac{28}{45} = \frac{28 \times 2}{45 \times 2} = \frac{56}{90} \]Now, subtract the fractions:
\[ \text{Difference} = \frac{75}{90} - \frac{56}{90} = \frac{75 - 56}{90} = \frac{19}{90} \]The distance is the absolute value of the difference:
\[ \text{Distance} = \left| \frac{19}{90} \right| = \frac{19}{90} \]The distance between points P and Q on the number line is \(\frac{19}{90}\).
Summary of Steps
- Identify the real numbers represented by points P and Q.
- Convert the repeating decimals to equivalent fractions.
- Calculate the difference between the two fractions.
- Take the absolute value of the difference to find the distance.
Comparing our result to the given options:
| Option | Value |
|---|---|
| 1 | \(\frac{21}{90}\) |
| 2 | \(\frac{19}{90}\) |
| 3 | \(\frac{21}{100}\) |
| 4 | \(\frac{56}{90}\) |
Our calculated distance, \(\frac{19}{90}\), matches Option 2.
Revision Table: Key Concepts Reviewed
| Concept | Description | Application in Problem |
|---|---|---|
| Distance on Number Line | The absolute difference between the coordinates of two points. For points \(a\) and \(b\), distance is \(|a-b|\). | Used to find the distance between points P (\(0.83\bar{}\)) and Q (\(0.62\bar{}\)). |
| Repeating Decimals | Decimals with a digit or sequence of digits that repeats infinitely. Represent rational numbers. | Given as the values for points P and Q. |
| Converting Repeating Decimals to Fractions | A method using algebraic manipulation (multiplying by powers of 10 and subtracting equations) to express repeating decimals as fractions of integers. | Applied to convert \(0.83\bar{}\) to \(\frac{5}{6}\) and \(0.62\bar{}\) to \(\frac{28}{45}\). |
| Subtracting Fractions | Requires a common denominator. Convert fractions to equivalent forms with the common denominator, then subtract the numerators. | Used to find the difference between \(\frac{5}{6}\) and \(\frac{28}{45}\). |
Additional Information: Rational and Real Numbers
Repeating decimals are a specific type of real number called rational numbers. A rational number is any number that can be expressed as the quotient or fraction \(\frac{p}{q}\) of two integers, where \(p\) is an integer and \(q\) is a non-zero integer. The decimal expansion of a rational number is either terminating (like 0.5 or 0.25) or repeating (like \(0.333...\) or \(0.142857142857...\)).
The number line is a visual representation of all real numbers. Every point on the number line corresponds to a unique real number, and every real number corresponds to a unique point on the number line. Real numbers include both rational numbers and irrational numbers (like \(\sqrt{2}\) or \(\pi\), whose decimal expansions are non-terminating and non-repeating).
In this problem, both \(0.83\bar{}\) and \(0.62\bar{}\) are rational numbers, and thus they are also real numbers represented by specific points on the number line.
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