Question

Two numbers are respectively \(20%\) and \(50%\) more than a third number. The ratio of the two numbers is:

• A. \(2:5\)
• B. \(3:5\)
• C. \(4:5\)
• D. \(6:7\)

Hint:

Whenever a quantity is “\(p%\) more/less” than a base, first convert to a multiplier \((1 \pm \tfrac{p{100)\) times the base.
In ratio questions, the base cancels out, letting you simplify cleanly to whole-number ratios.

Answer 1

The Correct Option is C

Step 1 (Let the base/third number be a variable).
Let the third number be \(x\).
“\(20%\) more than \(x\)” first number \(= x + \dfrac{20}{100}x = (1 + 0.20)x = 1.20x\).
“\(50%\) more than \(x\)” second number \(= x + \dfrac{50}{100}x = (1 + 0.50)x = 1.50x\).
Step 2 (Form the required ratio).
Required ratio \(= \text{first} : \text{second} = 1.20x : 1.50x\).
Step 3 (Cancel the common factor and convert to whole numbers).
Cancel the common factor \(x\) \(1.20 : 1.50\).
Write as fractions with whole numbers: \(1.20 : 1.50 = \dfrac{120}{100} : \dfrac{150}{100} = 120 : 150\).
Step 4 (Reduce the ratio to lowest terms).
\(\gcd(120,150) = 30\).
Divide both terms by \(30\) \(120:150 = \dfrac{120}{30} : \dfrac{150}{30} = 4 : 5\).
Step 5 (Conclude and map to option).
Hence, the ratio of the two numbers is \(4:5\), which matches Option 3.
\[ \boxed{4:5 \ \text{(Option (c)}} \]