Exercise 1.3 — Decimal Representation

What Does Exercise 1.3 Cover?

Exercise 1.3 from Chapter 1 — Rational Numbers brings together two important skills: converting decimals (including repeating decimals) into the p/q form, and applying rational number operations to solve word problems. These question types appear regularly in Class 8 board exams for CBSE, Telangana, and Andhra Pradesh, and they test both conceptual understanding and calculation accuracy.

Converting Terminating Decimals to p/q Form — Question 1

A terminating decimal is one that ends after a finite number of digits. To convert it, write the digits as the numerator and use a power of 10 matching the number of decimal places as the denominator, then simplify.

• 0.57 → 57/100 (2 decimal places → denominator 100)
• 0.176 → 176/1000 = 22/125 (simplify by dividing by 8)
• 1.00001 → 100001/100000 (5 decimal places → denominator 100000)
• 25.125 → 25125/1000 = 201/8 (simplify by dividing by 125)

Converting Repeating Decimals to p/q Form — Question 2

For recurring (repeating) decimals, use these standard rules based on which digits repeat:

• If all digits after the decimal repeat: numerator = repeating block, denominator = same number of 9s. Example: 0.9̄ = 9/9 = 1.
• If only some digits repeat: numerator = (full number − non-repeating part), denominator = 9s for repeating digits followed by 0s for non-repeating digits. Example: 0.5̄7̄ = 57/99 = 19/33.
• 0.7̄2̄9̄ → (729 − 7)/990 = 722/990 = 361/495.
• 12.28̄ → non-repeating part is 2, repeating part is 8: (28 − 2)/90 = 26/90 = 13/45 as the decimal part, so the full number is 12 and 13/45 = 553/45.

Algebraic Operations on Rational Numbers — Question 3

Question 3 asks you to find (x + y) ÷ (x − y) for given rational values. The approach is: compute x + y and x − y separately using cross-multiplication, then divide by multiplying by the reciprocal.

• x = 5/2, y = −3/4: x + y = 7/4, x − y = 13/4. Then (7/4) ÷ (13/4) = 7/13.
• x = 1/4, y = 3/2: x + y = 7/4, x − y = −5/4. Then (7/4) ÷ (−5/4) = −7/5.

Word Problems — Questions 4 to 10

These questions apply rational number arithmetic to real-life situations. Here are the key solutions and strategies:

• Q4 — Sum divided by product: Sum of −13/5 and 12/7 = −31/35. Product of −13/7 and −1/2 = 13/14. Dividing: −31/35 ÷ 13/14 = −62/65.
• Q5 — Finding an unknown number: If 2/5 of a number exceeds 1/7 of it by 36, set up 2x/5 − x/7 = 36. Solving gives 9x/35 = 36, so x = 140.
• Q6 — Rope problem: Two pieces of 2⅗ m and 3 3/10 m are cut from 11 m. Their combined length = 59/10 m. Remaining = 11 − 59/10 = 51/10 = 5 1/10 m.
• Q7 — Cost per metre: Cost of 7⅔ m is ₹12¾. Cost per metre = 51/4 ÷ 23/3 = 153/92 = ₹1 and 61/92.
• Q8 — Area of a park: Length 18⅗ m = 93/5 m, breadth 8⅔ m = 26/3 m. Area = 93/5 × 26/3 = 806/5 = 161⅕ sq. m.
• Q9 — Finding the divisor: −33/16 ÷ ? = −11/4. Required number = −33/16 ÷ (−11/4) = 3/4.
• Q10 — Cloth per trouser: 64 m for 36 trousers → 64/36 = 16/9 = 1 7/9 m per trouser.

Repeating Decimal — Bonus Question 11

For 0.363636… = 0.3̄6̄, using the recurring decimal rule: 36/99 = 4/11. But the question states the full number is 10.363636…, giving 10 and 4/11 = 114/11. So p = 114, q = 11, and p + q = 125.

Common Mistakes to Avoid

• When converting a mixed recurring decimal, do not put just 9s in the denominator — you need 0s for the non-repeating decimal digits too.
• In word problems, always convert mixed numbers to improper fractions before computing — working with mixed numbers directly leads to errors.
• Dividing by a fraction means multiplying by its reciprocal — do not invert the wrong fraction.

What This Exercise Prepares You For

The decimal-to-fraction conversions in this exercise are directly linked to the study of real numbers in Class 9, where you will distinguish between terminating, recurring, and non-recurring decimals. The word problem skills carry forward into linear equations and comparing quantities using proportion. For the foundational concepts behind this exercise, revisit Exercise 1.1 and the Introduction to Rational Numbers.