Question 22

We will use the given number cards and mathematical operations to get as close as possible to each target number. Each card can be used only once for each target.

Step 1 — Target 2,00,000

We want to reach 2,00,000. Let us try to combine the largest numbers first. We can add 1,50,000 and 70,000.

$1,50,000 + 70,000$

$= 2,20,000$

This is 20,000 more than our target. We need to subtract 20,000. Let us see if we can make 20,000 from the remaining cards. The remaining cards are 4000, 13000, 3000, 20, 5. We can multiply 4000 by 5.

$4000 \times 5$

$= 20,000$

Now, let us subtract this from 2,20,000.

$2,20,000 - 20,000$

$= 2,00,000$

This is exactly the target number. The cards used are 1,50,000, 70,000, 4000, and 5. Each card is used once.

$\boxed{1,50,000 + 70,000 - (4000 \times 5) = \mathbf{2,00,000}}$

Step 2 — Target 5,80,000

We want to reach 5,80,000. This is a large number. Let us try multiplying the largest cards by 5 or 20. If we multiply 70,000 by 5, we get 3,50,000.

$70,000 \times 5$

$= 3,50,000$

We need 5,80,000 - 3,50,000 = 2,30,000 more. The cards used are 70,000 and 5. The remaining cards are 4000, 13000, 3000, 150000, 20. We have 1,50,000. We need 2,30,000 - 1,50,000 = 80,000 more. Let us try to make 80,000 from the remaining cards: 4000, 13000, 3000, 20. We can multiply 4000 by 20.

$4000 \times 20$

$= 80,000$

Now, let us combine these parts.

$(70,000 \times 5) + 1,50,000 + (4000 \times 20)$

$= 3,50,000 + 1,50,000 + 80,000$

$= 5,00,000 + 80,000$

$= 5,80,000$

This is exactly the target number. The cards used are 70,000, 5, 1,50,000, 4000, and 20. Each card is used once.

$\boxed{(70,000 \times 5) + 1,50,000 + (4000 \times 20) = \mathbf{5,80,000}}$

Step 3 — Target 12,45,000

We want to reach 12,45,000. This is a very large number. Let us try multiplying a large card by 20. If we multiply 70,000 by 20, we get 14,00,000.

$70,000 \times 20$

$= 14,00,000$

This is more than our target. We need to subtract 14,00,000 - 12,45,000 = 1,55,000. The cards used are 70,000 and 20. The remaining cards are 4000, 13000, 3000, 150000, 5. We have 1,50,000. We need to subtract 1,55,000. If we subtract 1,50,000, we get 14,00,000 - 1,50,000 = 12,50,000.

$(70,000 \times 20) - 1,50,000$

$= 14,00,000 - 1,50,000$

$= 12,50,000$

This number is 12,50,000. Our target is 12,45,000. The difference is 12,50,000 - 12,45,000 = 5000. The cards used are 70,000, 20, and 1,50,000. The remaining cards are 4000, 13000, 3000, 5. We cannot make 5000 to subtract. Let us try another way to get close. Consider multiplying a sum of large numbers by 5. Let us add 1,50,000 and 70,000.

$1,50,000 + 70,000$

$= 2,20,000$

Now, multiply this sum by 5.

$2,20,000 \times 5$

$= 11,00,000$

We have 11,00,000. Our target is 12,45,000. We need 12,45,000 - 11,00,000 = 1,45,000 more. The cards used are 1,50,000, 70,000, and 5. The remaining cards are 4000, 13000, 3000, 20. Let us try to make 1,45,000 from these cards. Let us add 4000 and 3000.

$4000 + 3000$

$= 7000$

Now, multiply this sum by 20.

$7000 \times 20$

$= 1,40,000$

Now, let us combine these parts.

$(1,50,000 + 70,000) \times 5 + (4000 + 3000) \times 20$

$= 2,20,000 \times 5 + 7000 \times 20$

$= 11,00,000 + 1,40,000$

$= 12,40,000$

This number is 12,40,000. Our target is 12,45,000. The difference is 12,45,000 - 12,40,000 = 5000. The cards used are 1,50,000, 70,000, 5, 4000, 3000, and 20. The remaining card is 13000. We cannot make 5000 from 13000. Both 12,50,000 and 12,40,000 are 5000 away from 12,45,000. We can choose either one. Let us choose the one that uses more cards as it shows more operations.

$\boxed{(1,50,000 + 70,000) \times 5 + (4000 + 3000) \times 20 = \mathbf{12,40,000}}$

Step 4 — Target 20,90,800

We want to reach 20,90,800. This is a very large number. Let us try to combine large numbers and multiply by (20 + 5). Let us add 70,000 and 13,000.

$70,000 + 13,000$

$= 83,000$

Let us add 20 and 5.

$20 + 5$

$= 25$

Now, multiply these two sums.

$83,000 \times 25$

$= 20,75,000$

We have 20,75,000. Our target is 20,90,800. The difference is 20,90,800 - 20,75,000 = 15,800. The cards used are 70,000, 13,000, 20, and 5. The remaining cards are 4000, 3000, 150000. We need to add 15,800. Let us try to make a number close to 15,800 from the remaining cards. If we add 4000 and 3000, we get 7000.

$4000 + 3000$

$= 7000$

Now, let us add this to 20,75,000.

$$$$