This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here's the solution for Question 30: Question 30(a): Let the price of one pencil be x Ksh. i) Write an expression for the price of one pen. A pen costs Ksh 4 more than a pencil. Price of one pen = x + 4 Ksh. ii) Write an expression for the price of one pencil. Price of one pencil = x Ksh. Question 30(b): Kering purchased (2x-1) pens for Ksh 180. The price of one pen is (180)/(2x-1) Ksh. Naraya purchased (3x+1) pencils for Ksh 200. The price of one pencil is (200)/(3x+1) Ksh. From part (a), we know that the price of one pen is x+4 and the price of one pencil is x. Therefore, we can set up the equation based on the relationship: Price of one pen = Price of one pencil + 4. Step 1: Form the equation. (180)/(2x-1) = (200)/(3x+1) + 4 Step 2: Solve the equation for x. Multiply the entire equation by (2x-1)(3x+1) to eliminate denominators. 180(3x+1) = 200(2x-1) + 4(2x-1)(3x+1) 540x + 180 = 400x - 200 + 4(6x^2 + 2x - 3x - 1) 540x + 180 = 400x - 200 + 4(6x^2 - x - 1) 540x + 180 = 400x - 200 + 24x^2 - 4x - 4 Rearrange into a standard quadratic form ax^2 + bx + c = 0: 0 = 24x^2 + (400x - 4x - 540x) + (-200 - 4 - 180) 0 = 24x^2 - 144x - 384 Divide the entire equation by 24: (24x^2)/(24) - (144x)/(24) - (384)/(24) = 0 x^2 - 6x - 16 = 0 Step 3: Factor the quadratic equation. We need two numbers that multiply to -16 and add to -6. These numbers are -8 and 2. (x-8)(x+2) = 0 This gives two possible solutions for x: x-8 = 0 x = 8 x+2 = 0 x = -2 Step 4: Choose the valid value for x. Since x represents the price of a pencil, it must be a positive value. Also, the number of items purchased (2x-1) and (3x+1) must be positive. If x=-2, the price would be negative, and the number of pens would be 2(-2)-1 = -5, which is not possible. Therefore, the valid value is x = 8. The value of x is 8. Question 30(c): Step 1: Determine the original and new prices. From part (b), x=8. Original price of one pencil = x = 8 Ksh. Original price of one pen = x+4 = 8+4 = 12 Ksh. The price of a pen went up by 25%. New price of one pen = 12 × (1 + 0.25) = 12 × 1.25 = 15 Ksh. The price of a pencil remained unchanged. New price of one pencil = 8 Ksh. Step 2: Set up equations based on the given conditions. Let N_P be the number of pens bought by the school. Let N_Pc be the number of pencils bought by the school. Condition 1: The school spent the same amount of money on pens as on pencils. N_P × (New price of pen) = N_Pc × (New price of pencil) 15N_P = 8N_Pc (Equation 1) Condition 2: The total number of both pens and pencils bought was 46. N_P + N_Pc = 46 (Equation 2) Step 3: Solve the system of equations for N_P. From Equation 2, express N_Pc in terms of N_P: N_Pc = 46 - N_P Substitute this into Equation 1: 15N_P = 8(46 - N_P) 15N_P = 368 - 8N_P 15N_P + 8N_P = 368 23N_P = 368 N_P = (368)/(23) N_P = 16 The number of pens bought by the school is 16.