IMO Practice Test — Rational Numbers
6 Questions • 15 min • Olympiad level
15:00
Question 1 of 6
If \(\frac{a}{b} = \frac{2}{3}\) and \(\frac{b}{c} = \frac{4}{5}\), find \(\frac{a}{c}\).
8/15
15/8
6/12
5/6
Explanation: Multiply: a/c = (a/b)×(b/c) = (2/3)×(4/5) = 8/15
Question 2 of 6
Find the value of x: \(\frac{3}{4} + x = \frac{1}{2}\). Then find the multiplicative inverse of x.
-1/4, -4
-1/4, -1/4
1/4, 4
-1/4, 4
Explanation: x = 1/2 - 3/4 = 2/4 - 3/4 = -1/4; inverse = -4
Question 3 of 6
How many rational numbers lie between \(\frac{1}{3}\) and \(\frac{1}{2}\)?
0
1
2
Infinite
Explanation: Between any two rationals, there are infinitely many rationals
Question 4 of 6
If \(\frac{p}{q} = \frac{2}{3}\), then \(\frac{p+3}{q+3}\) is:
Equal to 2/3
Greater than 2/3
Less than 2/3
Cannot determine
Explanation: Take p=2,q=3: (2+3)/(3+3)=5/6≈0.833 > 0.667
Question 5 of 6
A number is such that when added to \(\frac{5}{8}\) gives \(\frac{3}{4}\). Find the number. Then find its reciprocal.
1/8, 8
1/8, 1/8
1/4, 4
-1/8, -8
Explanation: x = 3/4 - 5/8 = 6/8 - 5/8 = 1/8; reciprocal = 8
Question 6 of 6
Three numbers are in the ratio 2:3:4. If the sum of their reciprocals is \(\frac{13}{12}\), find the largest number.
2
3
4
6
Explanation: Numbers: 2x,3x,4x; 1/(2x)+1/(3x)+1/(4x)=13/12; LCM 12x: (6+4+3)/12x=13/12; 13/12x=13/12; x=1; Largest=4×1=4