Using the digits 2, 4, 5, 8, and 0 fill the boxes to get the:

(a) maximum product

(b) minimum product

(c) product greater than 150

(d) product nearest to 100

(e) product nearest to 5

Question

Using the digits 2, 4, 5, 8, and 0 fill the boxes to get the: (a) maximum product (b) minimum product (c) product greater than 150 (d) product nearest to 100 (e) product nearest to 5

Homi · Accepted Answer

Understand the Question

We need to arrange the five digits 0, 2, 4, 5, and 8 (each used exactly once) into the decimal form $\square.\square\square \times \square.\square$ (i.e., $N_1 = A.BC$ and $N_2 = D.E$ ).
To maximize or minimize: Place the largest/smallest available digits into the highest place-value positions (the whole number parts $A$ and $D$ ).
Evaluate specific combinations to answer each question.

(a) Using the digits 2, 4, 5, 8, and 0 fill the boxes to get the maximum product

Step 1 · Find Maximum Product

To get the maximum product, place the largest digits $8$ and $5$ as the integer parts. Diagram 1

Case 1: $A = 8, D = 5$ Using the remaining digits $0, 2, 4$ to make each factor as large as possible gives $N_1 = 8.42$ and $N_2 = 5.0$ . $8.42 \times 5.0 = 42.10$

Case 2: $A = 5, D = 8$ Using the remaining digits gives $N_1 = 5.42$ and $N_2 = 8.0$ . $5.42 \times 8.0 = 43.36$

Comparing the two products: $\text{Maximum product} = 43.36$

Answer

(a) $43.36$

(b) Using the digits 2, 4, 5, 8, and 0 fill the boxes to get the minimum product

Step 1 · Find Minimum Product

To get the minimum product, one of the integer parts must be $0$ . Diagram 2

Case 1: $N_1 = 0.BC$ Using the smallest remaining digits gives $N_1 = 0.24$ and $N_2 = 5.8$ . $0.24 \times 5.8 = 1.392$

Case 2: $N_2 = 0.E$ Using the smallest remaining digits gives $N_2 = 0.2$ and $N_1 = 4.58$ . $4.58 \times 0.2 = 0.916$

Comparing the two products: $\text{Minimum product} = 0.916$

Answer

(b) $0.916$

(c) Using the digits 2, 4, 5, 8, and 0 fill the boxes to get a product greater than 150

Step 1 · Check for Product Greater than 150

The maximum possible product obtainable using these digits is $43.36$ .

Since $43.36 < 150$ , it is not possible to obtain a product greater than $150$ .

Answer

(c) Not possible

(d) Using the digits 2, 4, 5, 8, and 0 fill the boxes to get the product nearest to 100

Step 1 · Find Product Nearest to 100

Since the maximum possible product is $43.36$ , all possible products are strictly less than $100$ .

Therefore, the product closest to $100$ is the largest possible product, which is $43.36$ .

Answer

(d) $43.36$

(e) Using the digits 2, 4, 5, 8, and 0 fill the boxes to get the product nearest to 5

Step 1 · Test Combinations Close to 5

Test possible combinations using all five digits:

For $N_1 = 0.54$ and $N_2 = 8.2$ : $0.54 \times 8.2 = 4.428 \implies |5 - 4.428| = 0.572$
For $N_1 = 0.82$ and $N_2 = 5.4$ : $0.82 \times 5.4 = 4.428 \implies |5 - 4.428| = 0.572$
For $N_1 = 0.52$ and $N_2 = 8.4$ : $0.52 \times 8.4 = 4.368 \implies |5 - 4.368| = 0.632$
For $N_1 = 5.42$ and $N_2 = 0.8$ : $5.42 \times 0.8 = 4.336 \implies |5 - 4.336| = 0.664$

The smallest difference from $5$ is $0.572$ , giving a product of $4.428$ .

Answer

(e) $4.428$

Common Mistakes

Digit Repetition/Omission: Using a digit more than once or missing a digit (e.g., trying $5.40 \times 0.8$ which repeats $0$ and misses $2$ ).
Minimization Strategy: Assuming $0.24 \times 5.8$ is smaller than $4.58 \times 0.2$ without checking both configurations of placing $0$ in the integer part.

Question 34

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