Test: Squares - Class 6 MCQ

The duplex method involves calculating the square of each digit and the product of each pair of digits multiplied by 2. Let's use the duplex method to find the square of 73.

The duplex method is an interesting way to calculate the square of a number. Let's use the duplex method to find the square of 88.

Step 1: Split the number into two digits

For 88, the digits are 8 and 8.

Step 2: Calculate the Duplex for each step

1. Duplex of the first digit (leftmost 8):

   8^2=64

2. Duplex of the first and second digit combined (8 and 8):

   2×(8×8)=2×64=128

Duplex of the second digit (rightmost 8):

8^2=64

Step 3: Combine the results

Start from the rightmost digit and combine the results, making sure to carry over any extra value to the next step.

• Rightmost digit (Step 3): 64. Write down 4, carry over 6.
• Middle step (Step 2): 128 + 6 (carry) = 134. Write down 4, carry over 13.
• Leftmost digit (Step 1): 64 + 13 (carry) = 77. Write down 77.

Step 4: Write down the final result

So, the square of 88 is 7744 using the duplex method.

Duplex combination or Dwanda yoga is term used in terms for squaring and multiplication,denoted as D.

1. Write the number with its digits separated: 4 4 94 \, 4 \, 9449

2. Calculate the duplex values:

   • Duplex of the first digit 4: 4×4=16
   • Duplex of the second digit 4: 4×4=16
   • Duplex of the third digit 9: 9×9=81
   • Duplex of the first and second digits: 4×4+4×4=16+16=32
   • Duplex of the second and third digits: 4×9+4×4=36+16=52
   • Duplex of the first and third digits: 4×9+4×4=36+16=52
   • Total duplex for the whole number:
     • For 4,4,9:
     • 16,32,16,52,81
     • Arrange these values in the appropriate places:
3. • The values you calculated should be arranged starting from the rightmost side to the left:
   • Starting with the last duplex 81, write it down.
   • Next, from 52, which represents the tens place, write 52 but note that it contributes to the tens place.
   • Next 32 contributes to the hundreds place.
   • Finally, 16 contributes to the thousands.

4. Add and carry if necessary:

   • The contributions will align into:
     • 81 (units),
     • 52 (tens) gives 5 carry,
     • 32+5=37 (hundreds),
     • 16 (thousands).

Now putting it all together, we get:

• From the right, 81
• Tens: 52 + 5 (carry) = 57 (so, 7 in the tens place, 5 carry over).
• Hundreds: 32 + 5 (carry) = 37 (so, 7 in the hundreds place, 3 carry over).
• Thousands: 16+3 (carry) = 19 (so, 9 in the thousands place, 1 carry over).

The result combines to 204,801.

Thus, the square of 449 is 201601.

So the final answers for the squares of the given numbers are:

• 205601 (which is 453²)
• 201601 (which is 449²)
• 203601 (which is 451²)
• 204601 (which is 452²).

To summarize:

• 449²=201601

Let's find the square of 976 using the duplex method:

1. Duplex of 9:
   9^2 = 81
   Write down 1 and carry over 8.

2. Duplex of 97:
   2×9×7=126
   Add the carry from the previous step:
   126+8=134
   Write down 4 and carry over 13.

3. Duplex of 976:
   2×(9×6)+7^2=2×54+49=108+49=157
   Add the carry from the previous step:
   157+13=170
   Write down 0 and carry over 17.

4. Duplex of 76:
   2×7×6=84
   Add the carry from the previous step:
   84+17=101
   Write down 1 and carry over 10.

5. Duplex of 6:
   6^2 = 36
   Add the carry from the previous step:
   36+10=46
   Write down 46.

So, the final result is 952576.

1. Split the Number: Break down 760 into its individual digits: 7, 6, and 0.

2. Calculate Duplexes:

   • D1 (single digits): The duplex of each individual digit is just its square. So, D1 for 7 is 72=4972=49, for 6 is 62=3662=36, and for 0 is 02=002=0.
   • D2 (pairs of digits): The duplex of a pair of digits is 2 times the product of those digits. For 76, it's 2×7×6=842×7×6=84, and for 60, it's 2×6×0=02×6×0=0. There's no pair involving the last 0 that gives a non-zero product since it's multiplied by 0.
   • D3 (triplets of digits): Since 760 is a 3-digit number, we don't have a full triplet involving different digits to consider for D3, as D3 typically involves the sum of products of three digits taken two at a time, multiplied by 2, plus the cube of the middle digit. However, in this method, we simplify by only considering adjacent pairs or single digits.

3. Apply the Duplex Method:

   • Write down the duplexes starting from the right (units place). The rightmost digit (D1 for the last digit) forms the units place of the answer.
   • Combine duplexes moving left, with each step involving addition and carrying over as needed, similar to traditional multiplication.

Calculation:

• The units place comes from the square of the last digit, which is 0 (02=002=0).
• Moving left, we add the duplexes. However, since the duplex involving 0 is 0, the next significant duplex comes from 62=3662=36, and the duplex of 76, which is 84.

Let's perform the calculation more straightforwardly, given the confusion that might arise from the duplex explanation above:

The square of 760 is 76027602. Let's calculate it directly:

760²=577600