Four numbers whose sum is 105, five jumbled spellings, five number series, and a three digit number from its digit sums

Understanding the topic

This question examines algebraic translation, vocabulary, pattern recognition and place value. The safest method is to turn each sentence into a precise equation before calculating. Two printed parts require caution: series (c)(iv) does not establish a unique mathematical rule, and part (d) uses “10th digit” where “tens digit” is intended.

(a) If the sum of four numbers is 105. When 03 is added to a number, twice of another number, five times of third number and fourth number become equal to each other. What are these numbers in ascending order?

Let the four numbers be A, B, C and D. The conditions give:

A + B + C + D = 105

and

A + 3 = 2B = 5C = D

Let their common value be k. Then:

• A = k - 3
• B = k/2
• C = k/5
• D = k

Substitute these expressions into the sum:

(k - 3) + k/2 + k/5 + k = 105

Multiply throughout by 10:

10k - 30 + 5k + 2k + 10k = 1050

27k = 1080

k = 40

Therefore:

• A = 40 - 3 = 37
• B = 40/2 = 20
• C = 40/5 = 8
• D = 40

Check: 37 + 20 + 8 + 40 = 105, while 37 + 3 = 2(20) = 5(8) = 40.

Answer in ascending order: 8, 20, 37, 40.

(b) Find out the correct word from the given jumbled spellings.

1. UCTREUTRS = STRUCTURE
2. LOVONAC = VOLCANO
3. CIHPROSTATAC = CATASTROPHIC
4. YNTIAUMH = HUMANITY
5. NNTHORER = NORTHERN

Each answer uses every supplied letter once. For example, CIHPROSTATAC rearranges into CATASTROPHIC, while NNTHORER rearranges into NORTHERN.

(c) Find the missing numbers in the series below

(i) 121, 11, 81, 9, ?, 7

The terms occur in square-root pairs:

• 121 = 11²
• 81 = 9²
• 49 = 7²

Missing number: 49.

(ii) 100, 50, 25, ?, 6.25

Each term is half the preceding term:

100 ÷ 2 = 50 ÷ 2 = 25 ÷ 2 = 12.5 ÷ 2 = 6.25

Missing number: 12.5.

(iii) 4, 9, 64, 125, 1296, ?

The bases rise from 2 to 7, while each exponent is used twice:

• 2² = 4 and 3² = 9
• 4³ = 64 and 5³ = 125
• 6⁴ = 1296 and 7⁴ = 2401

Missing number: 2401.

(iv) 2, 5, 12, 24, 48, ?

The printed terms do not determine one unique rule. The clearest intended continuation is repeated doubling from 12 onward:

12 × 2 = 24, 24 × 2 = 48, 48 × 2 = 96

Probable intended answer: 96. This should be presented with the ambiguity noted, rather than claiming that 96 is mathematically forced by the whole sequence.

(v) 44, 22, 66, 33, 132, ?

The operations alternate between division by 2 and multiplication by an increasing integer:

• 44 ÷ 2 = 22
• 22 × 3 = 66
• 66 ÷ 2 = 33
• 33 × 4 = 132
• 132 ÷ 2 = 66

Missing number: 66.

(d) If the sum of three digit number is 15 and sum of 10th and unit digit is 12. The difference of unit digit from 10th digit is equal to 02. What is the three digit number?

The printed “10th digit” means the tens digit. Let H, T and U represent the hundreds, tens and units digits.

The conditions are:

H + T + U = 15

T + U = 12

T - U = 2

Add the last two equations:

2T = 14

T = 7

Then:

U = 12 - 7 = 5

and:

H = 15 - 7 - 5 = 3

Place the digits in their stated positions: H = 3, T = 7 and U = 5.

Answer: 375.

Correction note: The answer 573 is inconsistent with the derived place values. It reverses the roles of the hundreds and units digits. The correct three-digit number is 375.