CBSE Class 7 Mathematics Fractions And Decimals Worksheet Set 09

Order of Operations (A)

Perform the operations in the correct order.

Question 1. \( \left(\frac{3}{2}\right)^{(-3) \times (-1)} \times \left( -4 - \left( -\frac{7}{6} - 3 \right) \right) \div \left( -\frac{7}{4} \right) \)
Answer: Follow the order of operations (BODMAS/PEMDAS) to solve step-by-step:

1. Calculate the exponent first:
\( (-3) \times (-1) = 3 \)
\( \implies \left(\frac{3}{2}\right)^3 = \frac{27}{8} \)

2. Solve inside the innermost parentheses:
\( -\frac{7}{6} - 3 = -\frac{7}{6} - \frac{18}{6} = -\frac{25}{6} \)

3. Solve the bracket:
\( -4 - \left(-\frac{25}{6}\right) = -4 + \frac{25}{6} = -\frac{24}{6} + \frac{25}{6} = \frac{1}{6} \)

4. Now, multiply and divide from left to right:
\( \frac{27}{8} \times \frac{1}{6} \div \left( -\frac{7}{4} \right) \)
\( \implies \frac{9}{16} \div \left( -\frac{7}{4} \right) \)
\( \implies \frac{9}{16} \times \left( -\frac{4}{7} \right) = -\frac{9}{28} \)

The final answer is \( -\frac{9}{28} \).
In simple words: First solve the exponent on the fraction. Then solve the numbers inside the brackets and finally multiply and divide from left to right.

Exam Tip: Be careful with signs. Remember that subtracting a negative number is the same as adding its positive value.

Question 2. \( \left( 1 \times \left( -\frac{1}{2} - 1 \right)^4 + (-11) \right) \times \frac{2}{3} \div \frac{1}{2} \)
Answer: Following the correct order of operations:

1. Solve inside the innermost parenthesis:
\( -\frac{1}{2} - 1 = -\frac{3}{2} \)

2. Evaluate the exponent:
\( \left(-\frac{3}{2}\right)^4 = \frac{81}{16} \)

3. Perform multiplication and then addition inside the large brackets:
\( 1 \times \frac{81}{16} = \frac{81}{16} \)
\( \implies \frac{81}{16} + (-11) = \frac{81}{16} - \frac{176}{16} = -\frac{95}{16} \)

4. Now, multiply and divide from left to right:
\( -\frac{95}{16} \times \frac{2}{3} \div \frac{1}{2} \)
\( \implies -\frac{95}{24} \div \frac{1}{2} \)
\( \implies -\frac{95}{24} \times 2 = -\frac{95}{12} = -7\frac{11}{12} \)

The final answer is \( -\frac{95}{12} \) (or \( -7\frac{11}{12} \)).
In simple words: Subtract 1 from \( -\frac{1}{2} \) first, then raise it to the 4th power. Add \( -11 \) and then complete the outer multiplication and division.

Exam Tip: Raise both the numerator and the denominator to the given power when evaluating exponents on fractions.

Question 3. \( 6 \div \left( \left( 3 + \left(-\frac{7}{3}\right) - (-1) \right)^3 - \left(-\frac{1}{2}\right) \right) \div 12 \)
Answer: Follow the order of operations step-by-step:

1. Solve the innermost brackets first:
\( 3 - \frac{7}{3} + 1 = 4 - \frac{7}{3} = \frac{12 - 7}{3} = \frac{5}{3} \)

2. Evaluate the exponent:
\( \left(\frac{5}{3}\right)^3 = \frac{125}{27} \)

3. Complete the subtraction inside the main bracket:
\( \frac{125}{27} - \left(-\frac{1}{2}\right) = \frac{125}{27} + \frac{1}{2} = \frac{250 + 27}{54} = \frac{277}{54} \)

4. Work through the division operations from left to right:
\( 6 \div \left(\frac{277}{54}\right) \div 12 \)
\( \implies 6 \times \frac{54}{277} \div 12 \)
\( \implies \frac{324}{277} \times \frac{1}{12} = \frac{27}{277} \)

The final answer is \( \frac{27}{277} \).
In simple words: Work out the nested brackets first to get a single fraction, and then divide from left to right.

Exam Tip: To divide by a fraction, multiply by its reciprocal. This is often the safest way to avoid calculation errors.

Question 4. \( \left( 1 + (-8) - (2 - (-1)) \times \left(-\frac{2}{3}\right) \right) \div \left( -\frac{1}{4} - (-1) \right) \)
Answer: Let's simplify the numerator-like bracket and denominator-like bracket separately:

1. Inside the left bracket:
\( 2 - (-1) = 3 \)
Next, multiply:
\( 3 \times \left(-\frac{2}{3}\right) = -2 \)
Now perform the remaining addition and subtraction:
\( 1 + (-8) - (-2) = 1 - 8 + 2 = -5 \)

2. Inside the right bracket:
\( -\frac{1}{4} - (-1) = -\frac{1}{4} + 1 = \frac{3}{4} \)

3. Finally, perform the division:
\( -5 \div \frac{3}{4} = -5 \times \frac{4}{3} = -\frac{20}{3} = -6\frac{2}{3} \)

The final answer is \( -\frac{20}{3} \) (or \( -6\frac{2}{3} \)).
In simple words: Simplify the two big groups of brackets first, then divide the first result by the second result.

Exam Tip: Breaking a large expression into distinct parts and solving them separately keeps your work neat and manageable.

Question 5. \( \left( (-1)^3 \times \frac{4}{3} \div \left( \frac{1}{3} \div \left( \left(-\frac{4}{3}\right) \div \left(-\frac{1}{2}\right) \right) \right) \right)^3 \)
Answer: Work from the innermost brackets outwards:

1. Solve the deepest bracket:
\( \left(-\frac{4}{3}\right) \div \left(-\frac{1}{2}\right) = -\frac{4}{3} \times (-2) = \frac{8}{3} \)

2. Solve the next level of bracket:
\( \frac{1}{3} \div \frac{8}{3} = \frac{1}{3} \times \frac{3}{8} = \frac{1}{8} \)

3. Solve the outer terms inside the main bracket (note that \( (-1)^3 = -1 \)):
\( -1 \times \frac{4}{3} \div \frac{1}{8} \)
\( \implies -\frac{4}{3} \times 8 = -\frac{32}{3} \)

4. Finally, apply the outer cube exponent:
\( \left( -\frac{32}{3} \right)^3 = -\frac{32768}{27} = -1213\frac{17}{27} \)

The final answer is \( -\frac{32768}{27} \) (or \( -1213\frac{17}{27} \)).
In simple words: Work outwards starting from the smallest inner bracket, then cube your final fraction.

Exam Tip: A negative number raised to an odd power (like 3) always results in a negative value.

Page 2

Order of Operations (A)

Perform the operations in the correct order.

Question 1. \( 6 - 6 \times (9 - 9) \div 1 \div (2 \times 6) \)
Answer: Follow the order of operations:

1. Solve inside brackets first:
\( 9 - 9 = 0 \)
\( 2 \times 6 = 12 \)
The expression becomes:
\( 6 - 6 \times 0 \div 1 \div 12 \)

2. Perform multiplication and division from left to right:
\( 6 \times 0 = 0 \)
\( 0 \div 1 = 0 \)
\( 0 \div 12 = 0 \)
The expression is now:
\( 6 - 0 = 6 \)

The final answer is 6.
In simple words: Since the bracket \( 9 - 9 \) equals 0, the entire multiplied term becomes 0, leaving just 6.

Exam Tip: Any multiplication chain containing a zero in it will simplify to zero. Spotting this early saves a lot of time.

Question 2. \( 8 \div 4 + 6 - 4 \div (3 - 2) \div 1 \)
Answer: Let's solve the terms:

1. Solve inside brackets:
\( 3 - 2 = 1 \)

2. Perform all divisions:
\( 8 \div 4 = 2 \)
\( 4 \div 1 \div 1 = 4 \)

3. Add and subtract from left to right:
\( 2 + 6 - 4 = 8 - 4 = 4 \)

The final answer is 4.
In simple words: Do the division parts first, then add and subtract to find the final value, which is 4.

Exam Tip: Never add or subtract numbers before completing all division and multiplication steps.

Question 3. \( 9 \times 2 \div (1 \div 1) - (3 + 9) \div 1 \)
Answer: Follow the order of operations:

1. Solve parentheses first:
\( 1 \div 1 = 1 \)
\( 3 + 9 = 12 \)

2. Perform multiplication and division from left to right:
\( 9 \times 2 = 18 \)
\( 18 \div 1 = 18 \)
\( 12 \div 1 = 12 \)

3. Perform subtraction:
\( 18 - 12 = 6 \)

The final answer is 6.
In simple words: Solve the brackets to get 1 and 12, do the division, and subtract the results to get 6.

Exam Tip: Keep parenthesis contents grouped as a single number before continuing with the rest of the equation.

Question 4. \( (4 + 2 - 6) \div 7 \div (2 \div (6 \div 3)) \)
Answer: Let's calculate from the inside out:

1. Solve inside parentheses:
\( 4 + 2 - 6 = 0 \)
\( 6 \div 3 = 2 \)

2. Substitute back into the expression:
\( 0 \div 7 \div (2 \div 2) \)
\( \implies 0 \div 7 \div 1 \)

3. Perform division:
\( 0 \div 7 = 0 \)
\( 0 \div 1 = 0 \)

The final answer is 0.
In simple words: The first bracket turns into 0. Since 0 divided by any non-zero number is 0, the final answer is 0.

Exam Tip: When zero is divided by any non-zero value, the result is always zero. This simplifies calculations instantly.

Question 5. \( 3 - (6 - 4) + 10 \div 10 + 8 - 4 \)
Answer: Solve using standard rules:

1. Solve parentheses first:
\( 6 - 4 = 2 \)

2. Perform division:
\( 10 \div 10 = 1 \)

3. Perform addition and subtraction from left to right:
\( 3 - 2 + 1 + 8 - 4 \)
\( \implies 1 + 1 + 8 - 4 \)
\( \implies 2 + 8 - 4 \)
\( \implies 10 - 4 = 6 \)

The final answer is 6.
In simple words: Simplify the bracket to 2, divide 10 by 10 to get 1, and then do the final addition and subtraction to get 6.

Exam Tip: When only addition and subtraction remain, always work strictly from left to right to get the correct result.

Question 6. \( 10 - 9 + 6 + 7 \times 1 \div 1 + 1 \)
Answer: Follow the order of operations:

1. Perform multiplication and division from left to right:
\( 7 \times 1 = 7 \)
\( 7 \div 1 = 7 \)

2. Perform addition and subtraction from left to right:
\( 10 - 9 + 6 + 7 + 1 \)
\( \implies 1 + 6 + 7 + 1 \)
\( \implies 7 + 7 + 1 = 15 \)

The final answer is 15.
In simple words: Multiply and divide first to get 7. Then, work out the addition and subtraction from left to right to get 15.

Exam Tip: Never mix up the order. Group multiplication and division together before moving on to addition.

Question 7. \( 1 \times 6 \div 3 \div 2 \times 9 \div (10 - 1) \)
Answer: Solve step-by-step:

1. Solve the parenthesis first:
\( 10 - 1 = 9 \)

2. Perform multiplication and division from left to right:
\( 1 \times 6 = 6 \)
\( 6 \div 3 = 2 \)
\( 2 \div 2 = 1 \)
\( 1 \times 9 = 9 \)
\( 9 \div 9 = 1 \)

The final answer is 1.
In simple words: Simplify the bracket to 9, then multiply and divide the remaining numbers from left to right to get 1.

Exam Tip: Since multiplication and division have equal priority, you must always evaluate them from left to right.

Question 8. \( 1 \times (6 - 5) \times 2 - 1 \times 8 \div 4 \)
Answer: Solve using the order of operations:

1. Solve parentheses first:
\( 6 - 5 = 1 \)

2. Perform multiplication and division from left to right:
\( 1 \times 1 \times 2 = 2 \)
\( 1 \times 8 \div 4 = 2 \)

3. Perform subtraction:
\( 2 - 2 = 0 \)

The final answer is 0.
In simple words: Solve the bracket to get 1. Do the multiplication and division on both sides to get 2 and 2, and then subtract them to get 0.

Exam Tip: Identify different terms separated by addition or subtraction to solve them as independent blocks first.

Question 9. \( 8 \div 2 \div (6 - 4) - (8 - (9 - 3)) \)
Answer: Solve starting with the innermost brackets:

1. Solve inside parentheses:
\( 6 - 4 = 2 \)
\( 9 - 3 = 6 \)

2. Substitute back to simplify the remaining brackets:
\( 8 \div 2 \div 2 - (8 - 6) \)
\( \implies 8 \div 2 \div 2 - 2 \)

3. Perform division from left to right:
\( 8 \div 2 = 4 \)
\( 4 \div 2 = 2 \)

4. Perform subtraction:
\( 2 - 2 = 0 \)

The final answer is 0.
In simple words: Work out the brackets first to get 2 and 2. Divide 8 by 2 and then by 2 to get 2. Finally, subtract 2 to get 0.

Exam Tip: When brackets are nested inside brackets, always solve the innermost one first and then work your way outwards.

Question 10. \( (1 - 1) \div (4 \div 2 + 6 + 4 - 2) \)
Answer: Solve inside both parentheses first:

1. Left bracket:
\( 1 - 1 = 0 \)

2. Right bracket:
\( 4 \div 2 + 6 + 4 - 2 = 2 + 6 + 4 - 2 = 10 \)

3. Perform division:
\( 0 \div 10 = 0 \)

The final answer is 0.
In simple words: The left side is 0, and the right side is 10. Since 0 divided by 10 is 0, the final answer is 0.

Exam Tip: Do not spend too much time calculating a complicated divisor bracket if you can easily see that the dividend evaluates to zero.