Arithmetic (Math) Solved Questions & Notes (2026) - Apex Rankers

Arithmetic mean:
(83+98)/2=181/2=90.5.
(Note: 91 is a common whole-int answer).

31/60=0.516... which is greater than 0.5. All others are less than 0.5.

n(n+1)/2=50(51)/2=1275.

91×91=8281.

Simplify each term using exponent rules:
When multiplying numbers with the same base, you add their exponents (\(2^a \cdot 2^b = 2^{a+b}\)).
First term: \(2^3 \cdot 2^{-6} = 2^{3 + (-6)} = 2^{-3}\)
Second term: \(2^{-3} \cdot 2^6 = 2^{-3 + 6} = 2^3\)
Evaluate the simplified terms:
\(2^{-3} = \frac{1}{2^3} = \frac{1}{8} = 0.125\)
\(2^3 = 8\)

Add the values together:
\(0.125+8=\mathbf{8.125}\)

Look for a perfect square factor of \(117\).
Since \(117 = 9 \times 13\), we can pull the square root of \(9\) outside the radical sign:
\(\sqrt{117}=\sqrt{9\times 13}=3\sqrt{13}\)
Substitute this back into the original expression:
\((3\sqrt{13})\times (3\sqrt{13})\)
Multiply the terms:
Multiply the coefficients outside the radical together, and multiply the square roots together:
\((3\times 3)\times (\sqrt{13}\times \sqrt{13})\)
\(9\times 13=\mathbf{117}\)

Identify the parts:
The ratio is 3 : 1, which means the numbers can be written as \(3x\) and \(1x\).
Find the sum: \(3x + 1x = 8 \implies 4x = 8 \implies x = 2\).
Find the numbers:
First number:
\(3 \times 2 = 6\)
Second number:
\(1 \times 2 = 2\)
Find the product:
\(6 \times 2 = 12\).

Here is the quick rule breakdown for divisibility by powers of 2:
Divisible by 2: The last 1 digit must be even (divisible by 2).
Divisible by 4: The last 2 digits must form a number divisible by 4.
Divisible by 8: The last 3 digits must form a number divisible by 8.

Find the value of 1/7 of the number:
If 5 parts out of 7 equal 1,025, then 1 part out of 7 is:
\(1,025\div 5=205\)
Find the value of 3/7 of the number:
Multiply the value of 1 part by 3:
\(205\times 3=615\)

Direct multiplication: 11.5×0.003=0.0345.

Weighted avg: (12×70+18×80)/30=(840+1440)/30=76.

If 9/12 are women, then 3/12 are men. 3/12=1/4=25%.

Note: Rs 60 / 0.60 = 100. The source key (d) suggests the budget was Rs 6.00 or a typo in unit.

Because the number of courses in both terms is equal, the combined average is simply the exact midpoint between the two term averages. The number exactly halfway between 75% and 85% is 80%

First, convert Rs. 100 to paise.
One rupee equals 100 paise.
So, Rs. 100 equals 10,000 paise (\(100 \times 100\)).
Write it as a fraction: \(\frac{50}{10000}\).
Simplify the fraction: \(\frac{1}{200}\).

Calculate the Total Price of the Books:The average price of the 3 books is Rs. 18.
\(\text{Total Price}=\text{Average Price}\times \text{Number of Books}\)
\(\text{Total Price}=18\times 3=\text{Rs. 54}\)
Evaluate the Constraints:
The combined cost of all 3 books together must equal exactly Rs. 54. Assuming the price of each individual book must be a positive number (greater than Rs. 0), no single book can cost as much as or more than the total amount of Rs. 54.
Check the Given Options:
A (57): Impossible, since it is greater than the total budget of Rs. 54.
B (56): Impossible, since it is greater than Rs. 54.
C (55): Impossible, since it is greater than Rs. 54.
D (52): Possible. If one book costs Rs. 52, the remaining two books could have a combined cost of Rs. 2 (e.g., Rs. 1 each).

To find the total number of boys in the row, we simply add the boys on both sides of Ajmal, plus Ajmal himself:
Boys to his left: 14
Boys to his right: 15
Ajmal himself: 1
\(\text{Total Boys}=14+15+1=\mathbf{30}\)

25 - 27 + 12 = 10

Using BODMAS: 60−12=48.

To find the score needed in the fifth subject, we can look at the total percentage points required:

Calculate the total points required for 5 subjects:
To maintain an overall average of 80% across 5 subjects:
\(5\times 80=400\text{ points}\)

Calculate the total points earned in the first 4 subjects:
With an average of 78% across 4 subjects:
\(4\times 78=312\text{ points}\)

Find the required score for the 5th subject:
Subtract the points already earned from the total points needed:
\(400-312=\mathbf{88\%}\)

LCM(7,8,9)=504.
504+5=509.

1. Simplify the Numerator
\(1.4+1\frac{2}{3}\div \frac{5}{3}\)
Convert the mixed fraction:
\(1\frac{2}{3} = \frac{5}{3}\)
Perform the division first:
\(\frac{5}{3} \div \frac{5}{3} = 1\)
Add to the decimal:
\(1.4 + 1 = \mathbf{2.4}\)

2. Simplify the Denominator
\(\frac{4}{3}\times 1\frac{4}{5}\div \frac{2}{5}\)
Convert the mixed fraction:
\(1\frac{4}{5} = \frac{9}{5}\)
Working left-to-right, perform the multiplication:
\(\frac{4}{3} \times \frac{9}{5} = \frac{36}{15} = \frac{12}{5}\)
Perform the division by multiplying by the reciprocal:
\(\frac{12}{5} \times \frac{5}{2} = \frac{12}{2} = \mathbf{6}\)

3. Divide Numerator by Denominator\(\frac{2.4}{6}=\mathbf{0.4}=\frac{2}{5}\)

Step 1: Simplify the First Part (Left Fraction)
Numerator:
\(12\frac{1}{3} + 4\frac{3}{15} \text{ of } \frac{1}{3}\)
Convert mixed numbers:
\(\frac{37}{3} + \frac{63}{15} \text{ of } \frac{1}{3} \rightarrow \frac{37}{3} + \frac{21}{5} \text{ of } \frac{1}{3}\)
Evaluate "of": \(\frac{21}{5} \times \frac{1}{3} = \frac{7}{5}\)
Add terms: \(\frac{37}{3} + \frac{7}{5} = \frac{185 + 21}{15} = \frac{206}{15}\)
Denominator: \(\frac{8}{9} \div \frac{2}{3} + \frac{5}{2} \text{ of } \frac{3}{5}\)
Evaluate "of": \(\frac{5}{2} \times \frac{3}{5} = \frac{3}{2}\)
Evaluate division: \(\frac{8}{9} \times \frac{3}{2} = \frac{4}{3}\)
Add terms: \(\frac{4}{3} + \frac{3}{2} = \frac{8 + 9}{6} = \frac{17}{6}\)
Combine Left Fraction:
\(\frac{206}{15}\div \frac{17}{6}=\frac{206}{15}\times \frac{6}{17}=\frac{412}{85}\)

Step 2: Simplify the Second Part (Right Fraction)
Numerator Bracket:
\(\left( \frac{1}{4} \div \frac{5}{4} + \frac{3}{2} \right)\)
Evaluate division:
\(\frac{1}{4} \times \frac{4}{5} = \frac{1}{5}\)
Add terms:
\(\frac{1}{5} + \frac{3}{2} = \frac{2 + 15}{10} = \frac{17}{10}\)
Full Numerator: \(\frac{4}{5} \text{ of } \frac{7}{8} \times \frac{17}{10}\)
Multiply out: \(\frac{4}{5} \times \frac{7}{8} \times \frac{17}{10} = \frac{7}{10} \times \frac{17}{10} = \frac{119}{100}\)
Denominator: \(1\frac{3}{4} + \frac{5}{8} - \frac{1}{8}\)
Convert mixed number: \(\frac{7}{4} + \frac{4}{8} = \frac{7}{4} + \frac{1}{2} = \frac{7 + 2}{4} = \frac{9}{4}\)
Combine Right Fraction:\(\frac{119}{100}\div \frac{9}{4}=\frac{119}{100}\times \frac{4}{9}=\frac{119}{225}\)

Step 3: Final Division
Now divide the simplified left fraction by the simplified right fraction:
\(\frac{412}{85}\div \frac{119}{225}=\frac{412}{85}\times \frac{225}{119}=\frac{412\times 45}{17\times 119}=\frac{18540}{2023}\)
Converting the improper fraction \(\frac{18540}{2023}\) into a mixed fraction:
\(18540 \div 2023 = 9\) with a remainder of \(333\)
Result: \(9\frac{333}{2023}\)

Step 1:
Deepest multiplication
\(\frac{2}{5}\times \frac{1}{3}=\frac{2}{15}\)
Step 2:
Next level up (division by the result)\(\frac{15}{\frac{2}{15}}=15\times \frac{15}{2}=\frac{225}{2}\)
Step 3:
Next level up\(\frac{3}{\frac{225}{2}}=3\times \frac{2}{225}=\frac{6}{225}=\frac{2}{75}\)
Step 4:
Perform the main middle division
\(\frac{5}{3}\div \frac{2}{75}=\frac{5}{3}\times \frac{75}{2}=\frac{5\times 25}{2}=\frac{125}{2}\)
Step 5: Solve the upper fraction block
\(\frac{3}{\frac{125}{2}}=3\times \frac{2}{125}=\frac{6}{125}\)
Step 6: Add to the main denominator base
\(2+\frac{6}{125}=\frac{250+6}{125}=\frac{256}{125}\)
Step 7:
Final top division\(\frac{15}{\frac{256}{125}}=15\times \frac{125}{256}=\frac{\mathbf{1875}}{\mathbf{256}}\)

0.0036+0.5007+4.1000=4.6043.