If the points P and Q represents real number \(0.7\bar 3\) and \(0.5\bar 6\) on the number line, then what is the distance between P and Q?
Calculating Distance Between Points on a Number Line
The question asks us to find the distance between two points, P and Q, on a number line. The points P and Q represent the real numbers \(0.7\bar 3\) and \(0.5\bar 6\) respectively. To find the distance between these points, we first need to convert the repeating decimals into fractions.
Converting Repeating Decimals to Fractions
Let's convert \(0.7\bar 3\) to a fraction:
- Let \(x = 0.7\bar 3 = 0.7333...\).
- Multiply by 10 to shift the decimal one place to the right: \(10x = 7.3333...\).
- Multiply by 100 to shift the decimal past the repeating part: \(100x = 73.3333...\).
- Subtract the equation for \(10x\) from the equation for \(100x\) to eliminate the repeating part:
\(100x - 10x = 73.3333... - 7.3333...\)
\(90x = 66\)
\(x = \frac{66}{90}\)
Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 6:
\(x = \frac{66 \div 6}{90 \div 6} = \frac{11}{15}\)
So, the point P represents the fraction \(\frac{11}{15}\).
Now, let's convert \(0.5\bar 6\) to a fraction:
- Let \(y = 0.5\bar 6 = 0.5666...\).
- Multiply by 10: \(10y = 5.6666...\).
- Multiply by 100: \(100y = 56.6666...\).
- Subtract the equation for \(10y\) from the equation for \(100y\):
\(100y - 10y = 56.6666... - 5.6666...\)
\(90y = 51\)
\(y = \frac{51}{90}\)
Simplify the fraction by dividing by the greatest common divisor, which is 3:
\(y = \frac{51 \div 3}{90 \div 3} = \frac{17}{30}\)
So, the point Q represents the fraction \(\frac{17}{30}\).
Calculating the Distance Between P and Q
The distance between two points on a number line is the absolute value of the difference between their coordinates. The coordinates are \(\frac{11}{15}\) for P and \(\frac{17}{30}\) for Q. The distance is \(|P - Q|\) or \(|Q - P|\). Let's calculate \(|P - Q|\):
Distance = \(|\frac{11}{15} - \frac{17}{30}|\)
To subtract these fractions, we need a common denominator. The least common multiple of 15 and 30 is 30. Convert \(\frac{11}{15}\) to an equivalent fraction with a denominator of 30:
\(\frac{11}{15} = \frac{11 \times 2}{15 \times 2} = \frac{22}{30}\)
Now substitute this back into the distance calculation:
Distance = \(|\frac{22}{30} - \frac{17}{30}|\)
Distance = \(|\frac{22 - 17}{30}|\)
Distance = \(|\frac{5}{30}|\)
The absolute value of \(\frac{5}{30}\) is \(\frac{5}{30}\). Simplify the fraction:
Distance = \(\frac{5 \div 5}{30 \div 5} = \frac{1}{6}\)
The distance between point P and point Q on the number line is \(\frac{1}{6}\).
Summary of Steps
- Convert the repeating decimal \(0.7\bar 3\) to the fraction \(\frac{11}{15}\).
- Convert the repeating decimal \(0.5\bar 6\) to the fraction \(\frac{17}{30}\).
- Calculate the absolute difference between the two fractions: \(|\frac{11}{15} - \frac{17}{30}|\).
- Find a common denominator and perform the subtraction: \(|\frac{22}{30} - \frac{17}{30}| = |\frac{5}{30}|\).
- Simplify the resulting fraction: \(\frac{5}{30} = \frac{1}{6}\).
Revision Table: Decimal to Fraction Conversion
| Decimal Form | Calculation Steps | Fraction Form |
|---|---|---|
| \(0.7\bar 3\) | \(x = 0.733...\) \(10x = 7.333...\) \(100x = 73.333...\) \(90x = 66\) \(x = \frac{66}{90}\) |
\(\frac{11}{15}\) |
| \(0.5\bar 6\) | \(y = 0.566...\) \(10y = 5.666...\) \(100y = 56.666...\) \(90y = 51\) \(y = \frac{51}{90}\) |
\(\frac{17}{30}\) |
Additional Information: Distance on Number Line
The distance between any two points, say 'a' and 'b', on a number line is given by the formula \(|a - b|\) or \(|b - a|\). The absolute value is used because distance is always a non-negative quantity. This concept is fundamental in understanding intervals and magnitudes on the real number line.
Similar Questions
If I = a 2 + b2 + c 2, where a and b are consecutive integers and c = ab, then I is
How many zeros are there in the product 1 50 × 2 49 × 3 48 × .... × 50 1 ?
A two-digit number is 9 more than four times of the number obtained by interchanging its digits. If the product of digits in the two-digit number is 8, then what is the number?
In a competitive examination, 250 students have registered. Out of these, 50 students have registered for Physics, 75 students for Mathematics and 35 students for both Mathematics and Physics. What is the number of students who have registered neither for Physics nor for Mathematics?
Consider the following statements:
1) If p is relatively prime to each of q and r, then p is relatively prime to the product qr.
2) If p divides the product qr and if p divides q, then p must divide r.
Which of the above statements is/are correct?If a, b and c are positive integers such that \(\dfrac{1}{a+\dfrac{1}{b+\dfrac{1}{c+\dfrac{1}{2}}}} =\dfrac{16}{23}\) , then what is the mean of a, b and c?
The inequality 3 N> N 3holds when
Let S be a set of fourteen natural numbers. The possible number of pairs (a, b), where a, b ∈ S and a ≠ b such that ab leaves remainder 1 when divided by 15, is
If m is the number of prime numbers between 0 and 50; and n is the number of prime numbers between 50 and 100, then what is (m – n) equal to?
If the sum of the digits of a number
10 n – 1, where n is a natural number, is equal to 3798, then what is the value of n?
Important Questions from Integers
Find the number of integers between $1$ and $150$ (inclusive) having $7$ as one of the digits but which are not divisible by $7$.
Integers are listed from 700 to 1000. In how many integers is the sum of the digits 10 ?
Using 2, 2, 3, 3, 3 as digits, how many distinct numbers greater than 30000 can be formed ?
Consider the following statements :
1. The sum of 5 consecutive integers can be 100.
2 The product of three consecutive natural numbers can be equal to their sum.
Which of the above statements is/are correct ?
The difference between a 2-digit number and the number obtained by interchanging the positions of the digits is 54.
Consider the following statements:
1. The sum of the two digits of the number can be determined only if the product of the two digits is known.
2. The difference between the two digits of the number can be determined.
Which of the above statements is/are correct?
