CBSE Class 6 Mathematics Revision Worksheet Set 22

Question 1. Find the least six digit number which can be divided by 16, 20, 24 and 28.
Answer: The least six-digit number is 100000.
First, find the LCM of 16, 20, 24, and 28.
Prime factorization:
\(16 = 2^4\)
\(20 = 2^2 \times 5\)
\(24 = 2^3 \times 3\)
\(28 = 2^2 \times 7\)
LCM = \(2^4 \times 3 \times 5 \times 7 = 16 \times 3 \times 5 \times 7 = 1680\)
Now, divide the least six-digit number (100000) by 1680 to find the remainder:
\(100000 \div 1680 = 59\) with a remainder of \(880\).
To find the least six-digit number divisible by 1680, subtract the remainder from 1680 and add it to 100000, or directly calculate:
\(100000 - 880 = 99120\) (This is a five-digit number)
Next multiple = \(99120 + 1680 = 100800\)
In simple words: Find the LCM of the given numbers, divide the smallest 6-digit number by this LCM, and add the difference between the divisor and the remainder to the 6-digit number.

Exam Tip: Always find the LCM first when dealing with numbers that need to be divided by a common set of divisors.

Question 2. Simplify \(100 - \{23 + (14 - 10 - 15)\}\)
Answer: Given expression: \(100 - \{23 + (14 - 10 - 15)\}\)
First, solve the innermost parentheses following the BODMAS rule:
\(14 - 10 - 15 = 4 - 15 = -11\)
Substitute this back into the expression:
\(100 - \{23 + (-11)\}\)
Now, solve inside the curly brackets:
\(23 + (-11) = 23 - 11 = 12\)
Finally, subtract from 100:
\(100 - 12 = 88\)
In simple words: Solve operations inside the innermost brackets first, then curly brackets, and finally subtract from 100.

Exam Tip: Be careful with signs when opening brackets, especially when subtracting a negative number or adding a negative term.

Question 3. Using distributive law, find the value of \(797 \times 571 - 797 \times 70 - 797\).
Answer: Rewrite the expression by identifying the common factor \(797\) (note that \(797\) can be written as \(797 \times 1\)):
\(= 797 \times 571 - 797 \times 70 - 797 \times 1\)
Take \(797\) common using the distributive law of multiplication over subtraction:
\(= 797 \times (571 - 70 - 1)\)
Simplify inside the parentheses:
\(571 - 70 - 1 = 571 - 71 = 500\)
Multiply by 797:
\(= 797 \times 500\)
\(= 398500\)
In simple words: Factor out the common number 797 and simplify the remaining terms inside the bracket before multiplying.

Exam Tip: Do not multiply directly; always look for common terms to apply the distributive property for faster calculations.

Question 4. Using division algorithm find the quotient when 36195 is divided by 154 and remainder is 5.
Answer: According to the division algorithm:
\(\text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}\)
Given:
Dividend = 36195
Divisor = 154
Remainder = 5
Substitute the values into the formula:
\(36195 = (154 \times \text{Quotient}) + 5\)
Subtract 5 from both sides:
\(36195 - 5 = 154 \times \text{Quotient}\)
\(36190 = 154 \times \text{Quotient}\)
Solve for Quotient:
\(\text{Quotient} = \frac{36190}{154}\)
\(\text{Quotient} = 235\)
In simple words: Subtract the remainder from the dividend, then divide the result by the divisor to get the quotient.

Exam Tip: Always verify your answer by multiplying the quotient by the divisor and adding the remainder.

Question 5. Arrange in descending order. \(\frac{2}{3}, \frac{4}{9}, \frac{5}{12}, \frac{1}{18}\)
Answer: To arrange fractions in descending order, first find the LCM of the denominators (3, 9, 12, 18).
Prime factorizations:
\(3 = 3\)
\(9 = 3^2\)
\(12 = 2^2 \times 3\)
\(18 = 2 \times 3^2\)
LCM = \(2^2 \times 3^2 = 4 \times 9 = 36\)
Now, convert each fraction to an equivalent fraction with a denominator of 36:
\(\frac{2}{3} = \frac{2 \times 12}{3 \times 12} = \frac{24}{36}\)
\(\frac{4}{9} = \frac{4 \times 4}{9 \times 4} = \frac{16}{36}\)
\(\frac{5}{12} = \frac{5 \times 3}{12 \times 3} = \frac{15}{36}\)
\(\frac{1}{18} = \frac{1 \times 2}{18 \times 2} = \frac{2}{36}\)
Comparing the numerators: \(24 > 16 > 15 > 2\)
Therefore, the descending order is: \(\frac{24}{36} > \frac{16}{36} > \frac{15}{36} > \frac{2}{36}\)
Or in original form: \(\frac{2}{3}, \frac{4}{9}, \frac{5}{12}, \frac{1}{18}\)
In simple words: Make all denominators the same by finding their LCM, then compare the numerators from largest to smallest.

Exam Tip: Descending order means arranging numbers from the greatest to the smallest.

Question 6. Add the difference of -4152 and 3056 to the difference of 1445 and -2947.
Answer: First difference (difference of -4152 and 3056):
\(= -4152 - 3056 = -7208\)
Second difference (difference of 1445 and -2947):
\(= 1445 - (-2947) = 1445 + 2947 = 4392\)
Now, add the two differences:
\(= -7208 + 4392 = -2816\)
In simple words: Calculate each difference separately, keeping track of negative signs, and then add the two results together.

Exam Tip: Be careful with subtraction involving negative numbers, as subtracting a negative is equivalent to addition.

Question 7. Simplify \(1 \frac{1}{5} \div \frac{3}{25} \text{ of } 1 \frac{1}{4} + 1 \frac{3}{10} \times \frac{5}{26}\)
Answer: Convert mixed fractions to improper fractions:
\(1 \frac{1}{5} = \frac{6}{5}\)
\(1 \frac{1}{4} = \frac{5}{4}\)
\(1 \frac{3}{10} = \frac{13}{10}\)
Substitute into the expression:
\(= \frac{6}{5} \div \left( \frac{3}{25} \text{ of } \frac{5}{4} \right) + \frac{13}{10} \times \frac{5}{26}\)
Perform "of" operation first (multiplication):
\(\frac{3}{25} \times \frac{5}{4} = \frac{3}{5} \times \frac{1}{4} = \frac{3}{20}\)
Now the expression is:
\(= \frac{6}{5} \div \frac{3}{20} + \frac{13}{10} \times \frac{5}{26}\)
Perform division by multiplying by the reciprocal:
\(\frac{6}{5} \div \frac{3}{20} = \frac{6}{5} \times \frac{20}{3} = 2 \times 4 = 8\)
Perform multiplication on the right part:
\(\frac{13}{10} \times \frac{5}{26} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\)
Add the two results:
\(= 8 + \frac{1}{4} = 8 \frac{1}{4} \text{ or } \frac{33}{4}\)
In simple words: Follow BODMAS by evaluating mixed fractions, solving "of" first, then division and multiplication, and finally addition.

Exam Tip: "Of" takes precedence over division and multiplication, so calculate it right after converting mixed fractions.

Question 8. i) LCM of two numbers is 4725. If the numbers are 525 and 945, find their HCF.
ii) Find the HCF of 384 and 512 by long division method.
Answer: Part (i):
We know the relation between HCF and LCM of two numbers:
\(\text{HCF} \times \text{LCM} = \text{Product of the two numbers}\)
\(\text{HCF} \times 4725 = 525 \times 945\)
\(\text{HCF} = \frac{525 \times 945}{4725}\)
Simplify the fraction:
\(\frac{4725}{525} = 9\)
\(\text{HCF} = \frac{945}{9} = 105\)

Part (ii):
Find HCF of 384 and 512 by long division method:
\(512 \div 384\): Quotient = 1, Remainder = \(512 - 384 = 128\)
\(384 \div 128\): Quotient = 3, Remainder = \(384 - (128 \times 3) = 0\)
Since the remainder becomes 0, the last divisor is the HCF.
HCF = 128
In simple words: For part (i), use the formula Product = HCF × LCM. For part (ii), divide the larger number by the smaller and repeat until the remainder is zero.

Exam Tip: The formula \(\text{HCF} \times \text{LCM} = a \times b\) is valid only for pairs of numbers.

Question 9. Find the value of \(\frac{(-45) \times (-30) - (-25) \times (-15)}{(-5)}\)
Answer: Numerator calculation:
First term: \((-45) \times (-30) = 1350\)
Second term: \((-25) \times (-15) = 375\)
Subtract the second term from the first term:
\(1350 - 375 = 975\)
Now, divide by the denominator (-5):
\(\frac{975}{-5} = -195\)
In simple words: Multiply the pairs in the numerator, subtract their products, and then divide the final numerator by -5.

Exam Tip: Remember that the product of two negative numbers is positive, and a positive number divided by a negative number gives a negative result.

Question 10. In the election candidate A got 45123 votes. Candidate B got 15305 votes more. But out of that 9843 votes were invalid. How many votes did Candidate B get? What was the total number of votes received by both the candidates?
Answer: Votes received by Candidate A = 45123
Candidate B got 15305 votes more than Candidate A.
Votes received by Candidate B = \(45123 + 15305 = 60428\)
Total number of votes received by both candidates (including valid and invalid as cast):
\(\text{Total votes} = \text{Votes of A} + \text{Votes of B}\)
\(\text{Total votes} = 45123 + 60428 = 105551\)
*(Note: The mention of invalid votes is additional data often used to find valid votes, but the question asks for the total votes received by both candidates).*
In simple words: Add 15305 to A's votes to get B's votes, then add A's and B's votes together to find the total.

Answer: Group the whole number parts and the fraction parts together:
\(= (30 + 18 - 12 - 25) + \left(\frac{2}{9} + \frac{1}{6} - \frac{2}{9} - \frac{5}{6}\right)\)
Simplify the whole numbers:
\(30 + 18 - 12 - 25 = 48 - 37 = 11\)
Simplify the fractions:
Notice that \(\frac{2}{9} - \frac{2}{9} = 0\), so the remaining fraction part is:
\(\frac{1}{6} - \frac{5}{6} = \frac{1 - 5}{6} = \frac{-4}{6} = -\frac{2}{3}\)
Combine the whole number and the fraction result:
\(= 11 + \left(-\frac{2}{3}\right) = 11 - \frac{2}{3}\)
Take LCM to subtract:
\(= \frac{33}{3} - \frac{2}{3} = \frac{31}{3} = 10 \frac{1}{3}\)
In simple words: Group whole numbers and fractions separately to make addition and subtraction easier.

Exam Tip: Grouping like terms (whole numbers with whole numbers, and fractions with matching denominators) saves time and reduces calculation errors.