Let's solve the problem step by step: --- ## **Step 1: Define the Blend Ratio** Let - \( x \) = proportion of Aggregate A - \( y \) = proportion of Aggregate B Since it's a binary blend: \[ x + y = 1 \] \[ y = 1 - x \] --- ## **Step 2: Write the Blending Formula** For each sieve size, the blended % passing is: \[ \text{Blended \% passing} = x \times (\text{\% passing of A}) + y \times (\text{\% passing of B}) \] --- ## **Step 3: Set Up Inequalities for Each Sieve** We'll use the recommended ranges for each size and set up inequalities: ### Example for 12.5 mm: \[ 80 \leq 90x + 100y \leq 100 \] Replace \( y = 1 - x \): \[ 80 \leq 90x + 100(1-x) \leq 100 \\ 80 \leq 90x + 100 - 100x \leq 100 \\ 80 \leq 100 - 10x \leq 100 \] \[ 80 \leq 100 - 10x \implies 10x \leq 20 \implies x \leq 2 \\ 100 - 10x \leq 100 \implies -10x \leq \implies x \geq \] So, \( \leq x \leq 2 \) (but \( x \leq 1 \) since \( x + y = 1 \)). We'll do this for each sieve. --- ## **Step 4: Calculate the Blended Limits for All Sieves** Let's do this systematically: ### **9.5 mm:** (Req: 70–90%) \[ 70 \leq 59x + 100y \leq 90 \] \[ 70 \leq 59x + 100(1-x) \leq 90 \] \[ 70 \leq 59x + 100 - 100x \leq 90 \\ 70 \leq 100 - 41x \leq 90 \] First inequality: \[ 70 \leq 100 - 41x \implies 41x \leq 30 \implies x \leq \frac{30}{41} \approx .732 \] Second: \[ 100 - 41x \leq 90 \implies -41x \leq -10 \implies x \geq \frac{10}{41} \approx .244 \] So: \[ .244 \leq x \leq .732 \] --- ### **4.75 mm:** (Req: 50–70%) \[ 50 \leq 16x + 96y \leq 70 \] \[ 50 \leq 16x + 96(1-x) \leq 70 \\ 50 \leq 16x + 96 - 96x \leq 70 \\ 50 \leq 96 - 80x \leq 70 \] First: \[ 50 \leq 96 - 80x \implies 80x \leq 46 \implies x \leq .575 \] Second: \[ 96 - 80x \leq 70 \implies -80x \leq -26 \implies x \geq .325 \] So: \[ .325 \leq x \leq .575 \] --- ### **2.36 mm:** (Req: 35–50%) \[ 35 \leq 3x + 75y \leq 50 \] \[ 35 \leq 3x + 75(1-x) \leq 50 \\ 35 \leq 3x + 75 - 75x \leq 50 \\ 35 \leq 75 - 72x \leq 50 \] First: \[ 35 \leq 75 - 72x \implies 72x \leq 40 \implies x \leq .556 \] Second: \[ 75 - 72x \leq 50 \implies -72x \leq -25 \implies x \geq .347 \] So: \[ .347 \leq x \leq .556 \] --- ### **.6 mm:** (Req: 18–29%) \[ 18 \leq x + 51y \leq 29 \] \[ 18 \leq 51(1-x) \leq 29 \] \[ 18 \leq 51 - 51x \leq 29 \] First: \[ 18 \leq 51 - 51x \implies 51x \leq 33 \implies x \leq .647 \] Second: \[ 51 - 51x \leq 29 \implies -51x \leq -22 \implies x \geq .431 \] So: \[ .431 \leq x \leq .647 \] --- ### **.3 mm:** (Req: 13–23%) \[ 13 \leq x + 30y \leq 23 \] \[ 13 \leq 30(1-x) \leq 23 \] \[ 13 \leq 30 - 30x \leq 23 \] First: \[ 13 \leq 30 - 30x \implies 30x \leq 17 \implies x \leq .567 \] Second: \[ 30 - 30x \leq 23 \implies -30x \leq -7 \implies x \geq .233 \] So: \[ .233 \leq x \leq .567 \] --- ### **.15 mm:** (Req: 8–16%) \[ 8 \leq x + 18y \leq 16 \] \[ 8 \leq 18(1-x) \leq 16 \] \[ 8 \leq 18 - 18x \leq 16 \] First: \[ 8 \leq 18 - 18x \implies 18x \leq 10 \implies x \leq .556 \] Second: \[ 18 - 18x \leq 16 \implies -18x \leq -2 \implies x \geq .111 \] So: \[ .111 \leq x \leq .556 \] --- ### **.075 mm:** (Req: 4–10%) \[ 4 \leq x + 9y \leq 10 \] \[ 4 \leq 9(1-x) \leq 10 \] \[ 4 \leq 9 - 9x \leq 10 \] First: \[ 4 \leq 9 - 9x \implies 9x \leq 5 \implies x \leq .556 \] Second: \[ 9 - 9x \leq 10 \implies -9x \leq 1 \implies x \geq \] So: \[ \leq x \leq .556 \] --- ## **Step 5: Find the Overlapping Range** List all the lower and upper bounds: - 12.5 mm: \( \leq x \leq 1\) - 9.5 mm: \(.244 \leq x \leq .732\) - 4.75 mm: \(.325 \leq x \leq .575\) - 2.36 mm: \(.347 \leq x \leq .556\) - .6 mm: \(.431 \leq x \leq .647\) - .3 mm: \(.233 \leq x \leq .567\) - .15 mm: \(.111 \leq x \leq .556\) - .075 mm: \( \leq x \leq .556\) **The valid range for \( x \) is the maximum of all lower bounds to the minimum of all upper bounds:** - Maximum lower bound: \( \max(.244, .325, .347, .431, .233, .111, ) = .431 \) - Minimum upper bound: \( \min(1, .732, .575, .556, .647, .567, .556, .556) = .556 \) --- ## **Step 6: Final Answer** \[ \boxed{.431 \leq x \leq .556} \] Where: - \( x \) = proportion of Aggregate A - \( y = 1 - x \) = proportion of Aggregate B **So, the blended ratio (by weight) of A : B should be between \(.431:.569\) and \(.556:.444\) to satisfy all requirements.** --- ### **Recommended Blend** A typical choice is the midpoint of the range: \[ x = \frac{.431 + .556}{2} = .493, \quad y = 1 - .493 = .507 \] **Final Answer:** \[ \boxed{\text{Blend Aggregate A : B in the ratio .49 : .51 (or any ratio where } .431 \leq x \leq .556\text{)}} \] --- ### **Summary Table** | Aggregate | Proportion (Min) | Proportion (Max) | |-----------|------------------|------------------| | A | .431 | .556 | | B | .569 | .444 | --- If you want to check for a specific ratio within this range, I can help calculate the blended gradation. Let me know!