Operations on Rational Numbers WordPress-Safe – No Scripts
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1) Concept Summary
Rational Number: A number of the form
mn
where m is any integer and n ≠ 0. Every integer a is rational because
\(a=\) a1.
A. Equivalent Fractions & Simplification
Multiplying (or dividing) numerator and denominator by the same non-zero number keeps the value same: 34 = 3×54×5 = 1520. Always reduce answers by dividing by the GCD.
B. Operations
- Addition/Subtraction: Use a common denominator (LCM), then operate on numerators.
512 + 718 = 1536 + 1436 = 2936.
- Multiplication: Multiply numerators & denominators; cancel common factors first.
1235 × 1418 = 235 × 143 = 28105 = 415.
- Division: Multiply by the reciprocal of the divisor.
89 ÷ 415 = 89 × 154 = 12036 = 103.
C. Multiplicative Inverse (Reciprocal)
For ab (non-zero), inverse is ba. Zero has no reciprocal.
D. Closure Properties
| Set | Add | Subtract | Multiply | Divide |
|---|---|---|---|---|
| Natural numbers | ✔ | ✘ (e.g., 7−10 = −3) | ✔ | ✘ (e.g., 3÷5 = 35) |
| Integers | ✔ | ✔ | ✔ | ✘ (e.g., 1÷2 = 12) |
| Rational numbers | ✔ | ✔ | ✔ | ✔ (except ÷0) |
E. Finding Numbers Between Two Rational Numbers
Make denominators equal, then pick numerators in between; or use the midpoint a+b2.
F. Decimal Forms
- Terminating iff (after reduction) the denominator has only 2’s and/or 5’s. Example: 134 = 3.25.
- Recurring otherwise. Example: 79 = 0.777…
G. Order of Operations (BODMAS)
Brackets → Of → Division & Multiplication (left-to-right) → Addition → Subtraction.
2) Practice Set 22 – Fully Worked
1. Carry out the following additions of rational numbers.
(i) 536 + 642
Simplify: 642 =
17. LCM(36, 7) = 252.
= 5×736×7 + 1×367×36 = 35252 + 36252 = 71252.
= 5×736×7 + 1×367×36 = 35252 + 36252 = 71252.
(ii) 123 + 245
= 53 +
145
= 2515 +
4215
= 6715
= 4715.
(iii) 1117 + 1319
= 11×19 + 13×1717×19
= 209 + 221323
= 430323.
(iv) 2311 + 1377
2311=2511=17577,
1377=8077.
Sum = 25577 = 32477.
Sum = 25577 = 32477.
2. Carry out the following subtractions involving rational numbers.
(i) 711 − 37
= 49 − 3377
= 1677.
(ii) 1336 − 240
240=120, LCM(36,20)=180.
= 65 − 9180 = 56180 = 1445.
= 65 − 9180 = 56180 = 1445.
(iii) 123 − 356
= 53 − 236
= 10 − 236
= −136
= −216.
(iv) 412 − 313
= 92 − 103
= 27 − 206
= 76.
3. Multiply the following rational numbers.
(i) 311 × 25
= 655.
(ii) 125 × 415
= 4875 = 1625.
(iii) (−89) × 34
= −2436 = −23.
(iv) 06 × 34
= 0.
4. Write the multiplicative inverse.
(i) 25 → 52
(ii) −38 → −83
(iii) −1739 → −3917
(iv) 7 → 17
(v) −713 = −223 → −322.
5. Carry out the divisions of rational numbers.
(i) 4012 ÷ 104
= 4012 × 410
= 160120
= 43.
(ii) −1011 ÷ −1110
= 1011 × 1011
= 100121.
(iii) −78 ÷ −36
= 78 × 63
= 4224
= 74.
(iv) 23 ÷ (−4)
= 23 × (−14)
= −212
= −16.
(v) 215 ÷ 53
215=115.
= 115 × 35 = 3325.
= 115 × 35 = 3325.
(vi) −513 ÷ 726
= −513 × 267
= −13091
= −107.
(vii) 911 ÷ (−8)
= 911 × (−18)
= −988.
(viii) 5 ÷ 25
= 5 × 52
= 252
= 1212.
3) Practice Set 23 – Write Three Rational Numbers Between
(i) Between 27 and 67
Same denominator: 37,
47,
57.
(ii) Between 45 and 23
Convert to /60: 4860, 4060.
Choose: 4160,
4260,
4360.
(iii) Between −23 and 45
Examples: −15, 0, 15.
(iv) Between 79 and −59
−49, −29, 0.
(v) Between −34 and +54
−12, 0, 1.
(vi) Between 78 and −53
−1, −12, 0.
(vii) Between 57 and 117
67, 1, 87.
(viii) Between 0 and −34
−14, −25, −12.
4) Practice Set 24 – Decimal Form (with steps)
(i) 134
= 3.25 (terminating since denominator is 2²).
(ii) −78
= −0.875 (terminating since denominator is 2³).
(iii) 735
= 7.6 (since 35=0.6).
(iv) 512
= 0.41666… = 0.416repeats.
(v) 227
= 3.142857142857… (cycle 142857 repeats).
(vi) 43
= 1.333… (3 repeats).
(vii) 79
= 0.777… (7 repeats).
5) Practice Set 25 – Simplify (BODMAS – full steps)
1) 50 × 5 ÷ 2 + 24
50×5=250 → 250÷2=125 → 125+24= 149.
2) (13 × 4) ÷ 2 − 26
(13×4)=52 → 52÷2=26 → 26−26= 0.
3) 140 ÷ [ (−11)×(−3) − (−42)÷14 − 1 ]
(−11)×(−3)=33, (−42)÷14=−3 → 33−(−3)−1=33+3−1=35 → 140÷35= 4.
4) { (220−140) + [ 10×9 + (−2×5) ] } − 100
(220−140)=80; 10×9=90, (−2×5)=−10 → [ ]=80 → { }=80+80=160 → 160−100= 60.
5) 35 + 38 ÷ 64
First division: 38 ÷ 64
= 38 × 46
= 1248
= 14.
Now add: 35 + 14 = 1220 + 520 = 1720.
Now add: 35 + 14 = 1220 + 520 = 1720.
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