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Mathbound · Contest practice

Mixed topics worksheet

Mixed difficulty

Name

Date

1.
An item costs $60$ . It is discounted $10\%$ , then $10\%$ tax is applied to the discounted price. How much is paid?
- A.61.40
- B.59.40
- C.57.90
- D.54
- E.66
2.
An item costs $52$ . After a $25\%$ discount (no tax), how much is paid?
- A.52
- B.40.00
- C.13
- D.39
- E.27
3.
What is $\operatorname{lcm}(8,36)$ ?
- A.72
- B.80
- C.288
- D.36
- E.4
4.
Circle radius $17$ . Chord length $30$ . Distance from center to chord?
- A.8
- B.2
- C.9
- D.17
- E.15
5.
What is $\operatorname{lcm}(12,24)$ ?
- A.36
- B.24
- C.0
- D.288
- E.12
6.
Solve for $x$ : $3x+10=28$ .
- A.5
- B.6
- C.28
- D.7
- E.10
7.
How many positive divisors does $25$ have?
- A.2
- B.3
- C.5
- D.4
- E.25
8.
Compute $3^{5}\bmod 13$ .
- A.3
- B.10
- C.11
- D.12
- E.9
9.
Lattice points strictly inside the right triangle with vertices $(0,0)$ , $(13,0)$ , $(0,7)$ ?
- A.37
- B.20
- C.45.5
- D.36
- E.35
10.
Lattice points strictly inside the right triangle with vertices $(0,0)$ , $(15,0)$ , $(0,8)$ ?
- A.60
- B.49
- C.48
- D.50
- E.23

Answer key · worked solutions

1. B. 59.40
Discount first, then tax. $60\cdot(1-10/100)\cdot(1+10/100)$ . $59.40$ .
2. D. 39
Multiply by $(1-d)$ . $52(1-25/100)$ . $39$ .
3. A. 72
$\operatorname{lcm}=ab/\gcd$ . $\gcd=4$ . $72$ .
4. A. 8
Right triangle: half-chord, radius, distance. $\sqrt{17^2-15^2}$ . $8$ .
5. B. 24
$\operatorname{lcm}(a,b)=ab/\gcd(a,b)$ . $\gcd=12$ . $24$ .
6. B. 6
Isolate $x$ . $3x=28-10$ . $6$ .
7. B. 3
Factor and use $\prod(e_i+1)$ . Count divisors of $25$ . $3$ .
8. E. 9
Successive squaring / cycles. Reduce powers modulo 13. $9$ .
9. D. 36
$\frac{(A-1)(B-1)}{2}$ for this triangle. Formula. $36$ .
10. B. 49
$\frac{(A-1)(B-1)}{2}$ for this triangle. Formula. $49$ .

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