Questions & explanations
1. If \(a\) and \(b\) are positive integers such that \(\frac{a}{b} < \frac{3}{4}\), which of the following must be true?
- \(\frac{a+1}{b+1} < \frac{3}{4}\)
- \(\frac{a+1}{b+1} > \frac{3}{4}\)
- \(\frac{a+1}{b+1} = \frac{3}{4}\)
- \(\frac{a}{b} < \frac{a+1}{b+1}\)
- \(\frac{a}{b} > \frac{a+1}{b+1}\)
Answer: \(\frac{a}{b} < \frac{a+1}{b+1}\)
Given \(\frac{a}{b} < \frac{3}{4}\), cross-multiply: \(4a < 3b\). Compare \(\frac{a}{b}\) and \(\frac{a+1}{b+1}\). Cross-multiply: \(a(b+1) < b(a+1) \Rightarrow ab + a < ab + b \Rightarrow a < b\). Since \(\frac{a}{b} < \frac{3}{4}\), it's possible that \(a < b\)? For positive integers, if \(a/b < 3/4\), then \(a < b\)? Not necessarily: e.g., \(a=3, b=4\) gives equality, but strict inequality requires \(a=2, b=3\) gives 2/3<3/4, and 2<3. if \(a/b < 1\), then \(a < b\). Since 3/4<1, \(a/b < 3/4 < 1\), so \(a < b\). Thus \(a < b\) holds, so \(\frac{a}{b} < \frac{a+1}{b+1}\). Option d is correct. Option e is false. Options a,b,c are not necessarily true; e.g., \(a=1,b=2\) gives \(\frac{2}{3} > \frac{3}{4}\)? 1/2=0.5, (1+1)/(2+1)=2/3≈0.666, still <3/4, so a holds? But must be true? For \(a=2,b=3\), (3/4)=0.75, (3/4)=0.75? (2+1)/(3+1)=3/4=0.75, so equality, not <. So a is not always true. b is false. c is not always true. d is always true because a<b.
2. Which of the following lists the fractions and decimals in order from least to greatest?
0.45, \(\frac{5}{11}\), 0.4545..., \(\frac{9}{20}\)
- \(\frac{9}{20}\), 0.45, \(\frac{5}{11}\), 0.4545...
- 0.45, \(\frac{9}{20}\), \(\frac{5}{11}\), 0.4545...
- \(\frac{5}{11}\), 0.45, \(\frac{9}{20}\), 0.4545...
- 0.45, \(\frac{5}{11}\), \(\frac{9}{20}\), 0.4545...
- \(\frac{9}{20}\), 0.45, 0.4545..., \(\frac{5}{11}\)
Answer: \(\frac{9}{20}\), 0.45, \(\frac{5}{11}\), 0.4545...
Convert each to decimal: 0.45 = 0.45; 5/11 ≈ 0.4545...; 0.4545... = 0.4545...; 9/20 = 0.45. So 9/20 = 0.45, then 5/11 = 0.4545..., and 0.4545... is the same as 5/11. 5/11 = 0.454545..., so 0.4545... (if terminating at 4 digits) is slightly less? But typically 0.4545... means repeating. So order: 0.45 (9/20) = 0.45, then 0.4545... (5/11) = 0.4545..., so least to greatest: 9/20, 0.45, 5/11, 0.4545... but 0.45 equals 9/20, so they are equal. However, the problem likely expects 9/20 = 0.45 exactly, and 5/11 = 0.4545..., so order: 9/20 and 0.45 are equal, then 5/11 and 0.4545... are equal. But options treat them as distinct. Option a: 9/20, 0.45, 5/11, 0.4545... is correct because 9/20 = 0.45, and 5/11 = 0.4545..., so they are in order.
3. If \(x\) and \(y\) are positive integers, is \(\frac{x}{y}\) a terminating decimal?
(1) \(x\) is a multiple of 2.
(2) \(y\) is a multiple of 3.
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- EACH statement ALONE is sufficient.
- Statements (1) and (2) TOGETHER are NOT sufficient.
Answer: Statements (1) and (2) TOGETHER are NOT sufficient.
A fraction in simplest form terminates if denominator has only prime factors 2 and 5. Statement (1): x being even doesn't guarantee denominator's factors. Statement (2): y multiple of 3 introduces factor 3, but fraction may simplify if x also multiple of 3. Together, if x is multiple of 2 and y multiple of 3, still unknown if fraction simplifies to remove factor 3. Example: x=2, y=3 gives 2/3 repeating; x=6, y=3 gives 2 terminating. Not sufficient.
4. Which of the following fractions has a decimal representation that terminates?
- \(\frac{7}{12}\)
- \(\frac{5}{14}\)
- \(\frac{3}{22}\)
- \(\frac{9}{30}\)
- \(\frac{11}{18}\)
Answer: \(\frac{9}{30}\)
A fraction in simplest form terminates if and only if the denominator has only prime factors 2 and 5. Simplify each: (a) 7/12, denominator 12=2^2*3, includes 3 → repeating. (b) 5/14, denominator 14=2*7, includes 7 → repeating. (c) 3/22, denominator 22=2*11, includes 11 → repeating. (d) 9/30 = 3/10 after simplifying (divide by 3), denominator 10=2*5 → terminates. (e) 11/18, denominator 18=2*3^2, includes 3 → repeating.
5. If \(x\) is a positive integer, is \(\frac{3}{x}\) greater than \(40\%\)?
(1) \(x < 7\)
(2) \(x > 5\)
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- EACH statement ALONE is sufficient.
- Statements (1) and (2) TOGETHER are NOT sufficient.
Answer: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
We need to check if 3/x > 0.4, i.e., 3/x > 2/5 → cross-multiplying (since x>0) gives 15 > 2x → x < 7.5. Since x is an integer, this is equivalent to x ≤ 7. Statement (1): x < 7 → x ≤ 6, so x ≤ 7 holds, thus 3/x > 0.4 is always true. Sufficient. Statement (2): x > 5 → x could be 6 (3/6=0.5>0.4) or 7 (3/7≈0.428>0.4) or 8 (3/8=0.375<0.4), so not always true. Insufficient. Hence statement (1) alone is sufficient.
6. Round 0.004987 to the nearest ten-thousandth.
- 0.0050
- 0.0049
- 0.005
- 0.00499
- 0.00500
Answer: 0.0050
The ten-thousandth place is the fourth decimal digit. 0.004987: digits: 0. 0 0 4 9 8 7. The ten-thousandth digit is 9 (fourth decimal). The next digit is 8 ≥ 5, so round up: 9 becomes 10, carry over: 0.0049 + 0.0001 = 0.0050. Express as 0.0050 to show rounding to ten-thousandth.
7. Which of the following fractions, when expressed in simplest form, yields a terminating decimal?
- \(\frac{2}{15}\)
- \(\frac{3}{20}\)
- \(\frac{5}{12}\)
- \(\frac{7}{18}\)
- \(\frac{1}{6}\)
Answer: \(\frac{3}{20}\)
A fraction in simplest form yields a terminating decimal if its denominator has only prime factors 2 and/or 5. \(\frac{3}{20}\) simplifies to \(\frac{3}{20}\) (already simplest), denominator 20 = 2^2 × 5, so it terminates.
8. Convert 5/11 to a percent. Which of the following is the correct percent representation?
- 45.45%
- 45.5%
- 45.4545%
- 45.45% (with a bar over the 45)
- 45%
Answer: 45.45% (with a bar over the 45)
5/11 = 0.454545..., a repeating decimal. To convert to percent, multiply by 100: 45.4545...% = 45.45% with a bar over the 45 to indicate repetition. Option d correctly shows the repeating decimal notation.
9. A company's revenue increased by 25% from Year 1 to Year 2, then decreased by 20% from Year 2 to Year 3. If the revenue in Year 1 was $240,000, what is the revenue in Year 3?
- $240,000
- $250,000
- $230,000
- $245,000
- $235,000
Answer: $240,000
Year 2 revenue = $240,000 × 1.25 = $300,000. Year 3 revenue = $300,000 × 0.80 = $240,000. The net change is 0% because a 25% increase followed by a 20% decrease results in a factor of 1.25 × 0.80 = 1.00.
10. Simplify: \(\frac{\frac{2x}{3y}}{\frac{4x^2}{9y^2}}\)
- \(\frac{3y}{2x}\)
- \(\frac{2x}{3y}\)
- \(\frac{3}{2}\)
- \(\frac{3y^2}{2x}\)
- \(\frac{2x^2}{3y^2}\)
Answer: \(\frac{3y}{2x}\)
Rewrite as division: \(\frac{2x}{3y} \div \frac{4x^2}{9y^2} = \frac{2x}{3y} \times \frac{9y^2}{4x^2} = \frac{2x \cdot 9y^2}{3y \cdot 4x^2} = \frac{18xy^2}{12x^2y} = \frac{3y}{2x}\) after canceling 6xy.
11. The decimal expansion of a certain fraction in simplest form is 0.181818... Which of the following is that fraction?
- \(\frac{2}{11}\)
- \(\frac{9}{50}\)
- \(\frac{18}{99}\)
- \(\frac{6}{33}\)
- \(\frac{3}{16}\)
Answer: \(\frac{2}{11}\)
0.181818... is a repeating decimal with repeating block '18'. Let x = 0.181818..., then 100x = 18.181818..., subtract: 99x = 18, so x = 18/99 = 2/11 in simplest form. Check: 2/11 = 0.181818...
12. What is the result of \( 2\frac{3}{4} + 1\frac{2}{3} \)?
- \( 3\frac{5}{7} \)
- \( 4\frac{5}{12} \)
- \( 4\frac{1}{4} \)
- \( 3\frac{5}{12} \)
- \( 4\frac{1}{3} \)
Answer: \( 4\frac{5}{12} \)
Convert to improper fractions: \( 2\frac{3}{4} = \frac{11}{4} \), \( 1\frac{2}{3} = \frac{5}{3} \). Common denominator 12: \( \frac{33}{12} + \frac{20}{12} = \frac{53}{12} = 4\frac{5}{12} \).