RBSE Solutions Class 8 Maths Chapter 1 Rational Numbers More Ques

🎯 Exam Tip: Review the basic properties of rational numbers, including inverses and the density property, to accurately match these concepts.

V. Very Short Answer Type Questions

Question 1. Define rational numbers.
Answer: A rational number is any number that can be written as a simple fraction \( \frac{p}{q} \). Here, 'p' and 'q' are whole numbers (integers), and 'q' cannot be zero. These numbers include fractions, integers, and terminating or repeating decimals.
In simple words: A rational number is like a fraction where the top and bottom parts are whole numbers, and the bottom number is not zero.

🎯 Exam Tip: Always remember the condition that the denominator 'q' must not be zero when defining rational numbers, as division by zero is not allowed.

Question 3. Find \( \frac {-2}{ 3 } \times \frac {4}{5} \)
Answer: To multiply these two fractions, we multiply the numerators (top numbers) together and the denominators (bottom numbers) together.
\( \left(\frac {-2}{ 3 } \right) \times \left( \frac { 4 }{ 5 } \right) \)
\( \implies \frac {-2 \times 4 }{ 3 \times 5 } \)
\( \implies \frac {-8 }{15} \)
Therefore, the product of \( \frac {-2}{3} \) and \( \frac {4}{5} \) is \( \frac {-8}{15} \).
In simple words: When multiplying fractions, just multiply the numbers on top and then multiply the numbers on the bottom. Keep the negative sign if there is one.

🎯 Exam Tip: Always check if you can simplify fractions before multiplying, as it can make the calculations easier. In this case, no simplification was possible.

Question 4. Write the additive inverse of \( -\frac {7}{19} \) and verify.
Answer: The additive inverse of \( -\frac {7}{19} \) is \( \frac {7}{19} \).
To verify, we add the number to its additive inverse, and the result should be zero.
\( -\frac {7}{19} + \frac {7}{19} \)
\( \implies \frac {-7 + 7}{19} \)
\( \implies \frac {0}{19} \)
\( \implies 0 \)
Since the sum is 0, the additive inverse is correctly found.
In simple words: The additive inverse of any number is simply that number with its sign changed. When you add a number and its additive inverse, they cancel each other out to make zero.

🎯 Exam Tip: Verification by addition is a crucial step to confirm that you have correctly identified the additive inverse.

Question 5. Find the mean of \( \frac {3}{8} \) and \( \frac {1}{2} \)
Answer: To find the mean (average) of two numbers, we add them together and then divide the sum by 2.
First, find the sum: \( \frac {3}{8} + \frac {1}{2} \)
Make the denominators the same by finding a common multiple, which is 8.
\( \implies \frac {3}{8} + \frac {1 \times 4}{2 \times 4} \)
\( \implies \frac {3}{8} + \frac {4}{8} \)
\( \implies \frac {3+4}{8} = \frac {7}{8} \)
Now, divide the sum by 2:
\( \implies \frac {7}{8} \div 2 \)
\( \implies \frac {7}{8} \times \frac {1}{2} \) (Dividing by a number is the same as multiplying by its reciprocal).
\( \implies \frac {7 \times 1}{8 \times 2} = \frac {7}{16} \)
The mean of \( \frac {3}{8} \) and \( \frac {1}{2} \) is \( \frac {7}{16} \).
In simple words: To find the middle value of two fractions, first add them up. Then, cut that sum in half. Make sure the fractions have the same bottom number before adding them.

🎯 Exam Tip: When finding the mean of fractions, ensure you correctly find a common denominator for addition and convert division into multiplication by the reciprocal.

Question 6. Is 0.6 multiplicative inverse of \( 1\frac {2}{3} \)? Why or why not?
Answer: Yes, 0.6 is the multiplicative inverse of \( 1\frac {2}{3} \).
To verify this, we multiply the two numbers, and if their product is 1, then they are multiplicative inverses.
First, convert \( 1\frac {2}{3} \) to an improper fraction: \( 1\frac {2}{3} = \frac {1 \times 3 + 2}{3} = \frac {5}{3} \).
Next, convert 0.6 to a fraction: \( 0.6 = \frac {6}{10} = \frac {3}{5} \).
Now, multiply the two fractions:
\( \frac {3}{5} \times \frac {5}{3} \)
\( \implies \frac {3 \times 5}{5 \times 3} \)
\( \implies \frac {15}{15} = 1 \)
Since their product is 1, 0.6 is indeed the multiplicative inverse of \( 1\frac {2}{3} \).
In simple words: To check if two numbers are multiplicative inverses, multiply them together. If the answer is 1, then they are. This is why \( \frac{3}{5} \) and \( \frac{5}{3} \) are inverses.

🎯 Exam Tip: Always convert all numbers to either fractions or decimals (preferably fractions) before checking for multiplicative inverse to avoid errors in calculation.

VI. Short Answer Type Questions

Question 1. Subtract \( \frac {-3}{8} \) from \( \frac {-5}{4} \)
Answer: To subtract \( \frac {-3}{8} \) from \( \frac {-5}{4} \), we set up the subtraction problem:
\( \frac {-5}{4} - \left( \frac {-3}{8} \right) \)
A minus sign followed by a negative sign becomes a plus sign:
\( \implies \frac {-5}{4} + \frac {3}{8} \)
Find a common denominator, which is 8. Multiply the numerator and denominator of \( \frac {-5}{4} \) by 2:
\( \implies \frac {-5 \times 2}{4 \times 2} + \frac {3}{8} \)
\( \implies \frac {-10}{8} + \frac {3}{8} \)
Now, add the numerators:
\( \implies \frac {-10 + 3}{8} \)
\( \implies \frac {-7}{8} \)
So, \( \frac {-3}{8} \) subtracted from \( \frac {-5}{4} \) is \( \frac {-7}{8} \).
In simple words: To subtract a negative fraction, it's like adding a positive one. Make sure both fractions have the same bottom number before you add or subtract the top numbers.

🎯 Exam Tip: Pay close attention to signs, especially when subtracting a negative number, as it changes to addition. Always use a common denominator for adding or subtracting fractions.

Question 2. Find ten rational number between \( \frac {-5}{6} \) and \( \frac {5}{8} \).
Answer: To find rational numbers between \( \frac {-5}{6} \) and \( \frac {5}{8} \), we first need to make their denominators common. The Least Common Multiple (LCM) of 6 and 8 is 24.
Convert \( \frac {-5}{6} \) to a fraction with a denominator of 24:
\( \frac {-5}{6} = \frac {-5 \times 4}{6 \times 4} = \frac {-20}{24} \)
Convert \( \frac {5}{8} \) to a fraction with a denominator of 24:
\( \frac {5}{8} = \frac {5 \times 3}{8 \times 3} = \frac {15}{24} \)
Now we need to find ten rational numbers between \( \frac {-20}{24} \) and \( \frac {15}{24} \). We can simply pick any ten fractions with a denominator of 24 that have numerators between -20 and 15.
Some examples of ten rational numbers are:
\( \frac {-19}{24}, \frac {-18}{24}, \frac {-17}{24}, \frac {-16}{24}, \frac {-15}{24}, \frac {-14}{24}, \frac {-13}{24}, \frac {-12}{24}, \frac {-11}{24}, \frac {-10}{24} \)
There are many other possible answers, as there are infinitely many rational numbers between any two given rational numbers.
In simple words: To find numbers between two fractions, change them so they have the same bottom number. Then, pick any fractions with that same bottom number and a top number that falls between the two original top numbers.

🎯 Exam Tip: When asked to find multiple rational numbers between two given numbers, increasing the common denominator by multiplying it by a factor (e.g., 2, 3, 4) can create more "space" between the numerators to choose from.

Question 3. Find the value of \( \frac {3}{7} + \left( \frac {-6}{11} \right) + \left( \frac {-8}{21} \right) + \left( \frac {5}{22} \right) \)
Answer: To find the sum of these rational numbers, we need to find a common denominator for all of them. The denominators are 7, 11, 21, and 22.
First, list the prime factors for each denominator:
7 = 7
11 = 11
21 = 3 x 7
22 = 2 x 11
The Least Common Multiple (LCM) will include the highest power of each prime factor present: \( 2 \times 3 \times 7 \times 11 = 462 \).
Now, convert each fraction to have a denominator of 462:
\( \frac {3}{7} = \frac {3 \times 66}{7 \times 66} = \frac {198}{462} \)
\( \frac {-6}{11} = \frac {-6 \times 42}{11 \times 42} = \frac {-252}{462} \)
\( \frac {-8}{21} = \frac {-8 \times 22}{21 \times 22} = \frac {-176}{462} \)
\( \frac {5}{22} = \frac {5 \times 21}{22 \times 21} = \frac {105}{462} \)
Now, add the converted fractions:
\( \frac {198}{462} + \left( \frac {-252}{462} \right) + \left( \frac {-176}{462} \right) + \left( \frac {105}{462} \right) \)
\( \implies \frac {198 - 252 - 176 + 105}{462} \)
Combine the positive terms and negative terms:
\( \implies \frac {(198 + 105) - (252 + 176)}{462} \)
\( \implies \frac {303 - 428}{462} \)
\( \implies \frac {-125}{462} \)
The value of the expression is \( \frac {-125}{462} \).
In simple words: To add or subtract many fractions, first find the smallest common bottom number for all of them. Change each fraction to use this new bottom number, then add or subtract the top numbers.

🎯 Exam Tip: Always calculate the LCM of all denominators accurately before adding or subtracting multiple fractions to ensure correct conversions and a streamlined calculation.

Question 4. Product of two rational number is \( \frac {-28}{81} \). If one rational number is \( \frac {14}{27} \) then find the other.
Answer: Let the other rational number be 'x'.
According to the problem, the product of two rational numbers is \( \frac {-28}{81} \).
One rational number is \( \frac {14}{27} \).
So, we can write the equation:
\( \frac {14}{27} \times x = \frac {-28}{81} \)
To find 'x', we need to divide the product by the known rational number:
\( x = \frac {-28}{81} \div \frac {14}{27} \)
Dividing by a fraction is the same as multiplying by its reciprocal:
\( x = \frac {-28}{81} \times \frac {27}{14} \)
Now, simplify by canceling common factors:
(Note that \( -28 = -2 \times 14 \) and \( 81 = 3 \times 27 \))
\( x = \frac {-2 \times 14}{3 \times 27} \times \frac {27}{14} \)
Cancel out 14 and 27:
\( x = \frac {-2}{3} \)
Therefore, the other rational number is \( \frac {-2}{3} \).
In simple words: If you know the total when two numbers are multiplied, and you know one of the numbers, you can find the other by dividing the total by the known number. Remember to flip the fraction when you divide.

🎯 Exam Tip: When dividing by a fraction, always convert it to multiplication by the reciprocal. Look for opportunities to simplify by canceling common factors before multiplying.

Question 5. Find the value of \( (x + y) \div (x - y) \) If \( x = \frac {5}{4}, y = -\frac {1}{3} \)
Answer: We are given \( x = \frac {5}{4} \) and \( y = -\frac {1}{3} \). We need to find the value of \( (x + y) \div (x - y) \).
First, calculate \( (x + y) \):
\( x + y = \frac {5}{4} + \left( -\frac {1}{3} \right) \)
\( \implies \frac {5}{4} - \frac {1}{3} \)
Find a common denominator for 4 and 3, which is 12:
\( \implies \frac {5 \times 3}{4 \times 3} - \frac {1 \times 4}{3 \times 4} \)
\( \implies \frac {15}{12} - \frac {4}{12} \)
\( \implies \frac {15 - 4}{12} = \frac {11}{12} \)
Next, calculate \( (x - y) \):
\( x - y = \frac {5}{4} - \left( -\frac {1}{3} \right) \)
\( \implies \frac {5}{4} + \frac {1}{3} \)
Find a common denominator, which is 12:
\( \implies \frac {5 \times 3}{4 \times 3} + \frac {1 \times 4}{3 \times 4} \)
\( \implies \frac {15}{12} + \frac {4}{12} \)
\( \implies \frac {15 + 4}{12} = \frac {19}{12} \)
Finally, calculate \( (x + y) \div (x - y) \):
\( \frac {11}{12} \div \frac {19}{12} \)
\( \implies \frac {11}{12} \times \frac {12}{19} \) (Multiply by the reciprocal)
Cancel out the common factor 12:
\( \implies \frac {11}{19} \)
The value of \( (x + y) \div (x - y) \) is \( \frac {11}{19} \).
In simple words: First, add the two given fractions, and then subtract them. After that, divide the sum by the difference. Remember to change subtraction of a negative to addition.

🎯 Exam Tip: Break down complex problems into smaller, manageable steps. Calculate \( (x+y) \) and \( (x-y) \) separately first, ensuring correct handling of signs and common denominators, before performing the final division.

Question 6. By what number should we multiply \( \frac {3}{-14} \), so that the product may be \( \frac {5}{12} \).
Answer: Let the unknown number be 'k'.
We are looking for a number 'k' such that when multiplied by \( \frac {3}{-14} \), the product is \( \frac {5}{12} \).
So, we have the equation:
\( \frac {3}{-14} \times k = \frac {5}{12} \)
To find 'k', divide \( \frac {5}{12} \) by \( \frac {3}{-14} \):
\( k = \frac {5}{12} \div \frac {3}{-14} \)
Remember that dividing by a fraction is the same as multiplying by its reciprocal:
\( k = \frac {5}{12} \times \frac {-14}{3} \)
Now, multiply the numerators and denominators. We can also simplify by dividing common factors (14 and 12 can both be divided by 2):
\( k = \frac {5 \times (-14)}{12 \times 3} \)
\( k = \frac {5 \times (-7 \times 2)}{6 \times 2 \times 3} \)
Cancel out the common factor 2:
\( k = \frac {5 \times (-7)}{6 \times 3} \)
\( k = \frac {-35}{18} \)
The number is \( \frac {-35}{18} \).
In simple words: To find a missing number in a multiplication problem, divide the product by the known number. Remember to flip the fraction you are dividing by.

🎯 Exam Tip: Always simplify fractions before multiplying to work with smaller numbers and reduce the chance of calculation errors.

Question 7. Subtract the sum of \( \frac {5}{14} \) and \( \frac {-4}{7} \) from the sum of \( \frac {9}{14} \) and \( \frac {23}{14} \)
Answer: We need to calculate two sums first and then subtract the first sum from the second sum.
**Step 1: Calculate the first sum (Sum 1)**
Sum 1 = \( \frac {5}{14} + \left( \frac {-4}{7} \right) \)
Find a common denominator for 14 and 7, which is 14:
Sum 1 = \( \frac {5}{14} + \frac {-4 \times 2}{7 \times 2} \)
Sum 1 = \( \frac {5}{14} + \frac {-8}{14} \)
Sum 1 = \( \frac {5 - 8}{14} = \frac {-3}{14} \)
**Step 2: Calculate the second sum (Sum 2)**
Sum 2 = \( \frac {9}{14} + \frac {23}{14} \)
Since the denominators are already the same, just add the numerators:
Sum 2 = \( \frac {9 + 23}{14} \)
Sum 2 = \( \frac {32}{14} \)
**Step 3: Subtract Sum 1 from Sum 2**
Result = Sum 2 - Sum 1
Result = \( \frac {32}{14} - \left( \frac {-3}{14} \right) \)
A minus sign followed by a negative sign becomes a plus sign:
Result = \( \frac {32}{14} + \frac {3}{14} \)
Add the numerators:
Result = \( \frac {32 + 3}{14} \)
Result = \( \frac {35}{14} \)
This fraction can be simplified by dividing both numerator and denominator by their greatest common divisor, which is 7:
Result = \( \frac {35 \div 7}{14 \div 7} = \frac {5}{2} \)
The final value is \( \frac {5}{2} \).
In simple words: First, add the first pair of fractions. Then, add the second pair of fractions. Finally, take the result of the second sum and subtract the result of the first sum from it. Always make sure fractions have the same bottom number before adding or subtracting.

🎯 Exam Tip: When dealing with multiple operations on fractions, always calculate sums or differences within parentheses first, and ensure all fractions have common denominators for addition/subtraction.

Question 8. \( 3\frac {1}{2} + \left( -1\frac {1}{2} \right) - \left( -4\frac {1}{4} \right) + \left( 2\frac {2}{5} \right) \)
Answer: First, convert all mixed numbers to improper fractions and simplify the signs.
\( 3\frac {1}{2} = \frac {3 \times 2 + 1}{2} = \frac {7}{2} \)
\( -1\frac {1}{2} = -\frac {1 \times 2 + 1}{2} = -\frac {3}{2} \)
\( -4\frac {1}{4} = -\frac {4 \times 4 + 1}{4} = -\frac {17}{4} \)
\( 2\frac {2}{5} = \frac {2 \times 5 + 2}{5} = \frac {12}{5} \)
Now substitute these into the expression:
\( \frac {7}{2} + \left( -\frac {3}{2} \right) - \left( -\frac {17}{4} \right) + \frac {12}{5} \)
Simplify the signs: a plus with a minus becomes minus, and a minus with a minus becomes plus.
\( \implies \frac {7}{2} - \frac {3}{2} + \frac {17}{4} + \frac {12}{5} \)
Now, find the Least Common Multiple (LCM) of the denominators 2, 4, and 5. The LCM of 2, 4, and 5 is 20.
Convert each fraction to have a denominator of 20:
\( \frac {7}{2} = \frac {7 \times 10}{2 \times 10} = \frac {70}{20} \)
\( \frac {3}{2} = \frac {3 \times 10}{2 \times 10} = \frac {30}{20} \)
\( \frac {17}{4} = \frac {17 \times 5}{4 \times 5} = \frac {85}{20} \)
\( \frac {12}{5} = \frac {12 \times 4}{5 \times 4} = \frac {48}{20} \)
Substitute these back into the expression:
\( \frac {70}{20} - \frac {30}{20} + \frac {85}{20} + \frac {48}{20} \)
Combine the numerators:
\( \implies \frac {70 - 30 + 85 + 48}{20} \)
\( \implies \frac {40 + 85 + 48}{20} \)
\( \implies \frac {125 + 48}{20} \)
\( \implies \frac {173}{20} \)
This can also be expressed as a decimal or a mixed number:
\( \frac {173}{20} = 8\frac {13}{20} = 8.65 \)
The final value is \( \frac {173}{20} \) or 8.65.
In simple words: First, change all mixed numbers into simple fractions. Then, deal with the plus and minus signs so there is only one sign between each fraction. Next, find a common bottom number for all fractions and change them. Finally, add and subtract the top numbers to get the answer.

🎯 Exam Tip: Pay very close attention to changing mixed numbers to improper fractions and simplifying double negative signs, as these are common sources of errors.