Practice Maths

Substitution and Formulas — Solutions

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  1. Substitute into formulas

    1. A = lw, l = 12, w = 7:
    2. P = 2(l + w), l = 9, w = 5:
    3. A = ½bh, b = 8, h = 11:
    4. C = 2πr, r = 6:
    5. A = πr², r = 5:
    6. V = lwh, l = 4, w = 3, h = 7:
    7. V = πr²h, r = 2, h = 9:
    8. v = u + at, u = 0, a = 3, t = 8:
    9. I = Prt ÷ 100, P = 2000, r = 5, t = 3:
    10. s = d ÷ t, d = 360, t = 4:

  2. Evaluate algebraic expressions

    1. 3x + 7, x = 4:
    2. 2a² − 5, a = 3:
    3. 4m − 2n, m = −2, n = 3:
    4. p² + 2pq, p = 5, q = −1:
    5. (x + y) ÷ (x − y), x = 9, y = 3:
    6. 3c² − 4c + 1, c = −2:
    7. √(a² + b²), a = 3, b = 4:
    8. (2x + 1)/(x − 1), x = 4:

  3. Work backwards

    1. A = lw, A = 48, w = 6, find l:
    2. v = u + at, v = 35, u = 5, t = 6, find a:
    3. A = πr², A = 78.54, find r:
    4. I = Prt ÷ 100, I = 300, P = 1000, r = 5, find t:
    5. V = lwh, V = 120, l = 5, h = 4, find w:

  4. Formula applications

    1. P = 34, l = 10, find w:
    2. v = 80, t = 2.5, d = vt:
    3. I = 240, P = 1200, t = 2, find r:
    4. E = ½mv², m = 5, v = 12:

  5. Multi-formula problem solving

    1. Cylindrical tank, r = 1.5, h = 3:
      1. Volume:
      2. Days to empty:
    2. h = −5t² + 20t + 1:
      1. t = 1:
      2. t = 3:
      3. t = 4:
    3. A = 4πr²:
      1. r = 7:
      2. A = 200.96, find r:
    4. Simple interest comparison:

  6. Physics formulas

    1. F = ma, m = 12, a = 3.5:
    2. v² = u² + 2as, u = 0, a = 10, s = 45:
    3. p = mv, m = 0.5, v = 60:
    4. E = mgh, m = 3, g = 9.8, h = 15:
    5. s = ut + ½at², u = 5, t = 4, a = 2:

  7. Negative and fractional substitution

    1. T = −2x² + 6x + 8, x = −1:
    2. R = (a − b)/(a + b), a = −3, b = −5:
    3. C = &frac59;(F − 32), F = 14:
    4. y = 3x² − 2x + 1, x = −½:
    5. S = n⁄2(a + l), n = 12, a = 3, l = 47:

  8. Choose and apply the right formula

    1. Ball accelerating from rest:
    2. Train speed:
    3. Spherical balloon r = 4:
    4. Simple interest P = 8000, r = 3.5, t = 4:
    5. Pizza diameter 30 cm, so r = 15:

  9. Compare and decide

    1. Pools r = 3 and r = 5:
    2. Kinetic energy:
    3. Simple interest comparison:
    4. Rocket h = 80t − 5t²:

  10. Real-world formulas

    1. E = mc², m = 2 × 10³, c = 3 × 10&sup8;:
    2. BMI = m/h², m = 72, h = 1.75:
    3. A = P(1 + r)³, P = 1000, r = 0.05:
    4. Ohm’s Law V = IR: