Calculate the value of (-7) + (+3) - (-4).
- -8
- -14
- 8
- 0 Correct answer.
Explanation
Subtracting a negative number becomes addition, so -(-4) = +4. The expression becomes -7 + 3 + 4. Computing left to right: -7 + 3 = -4, then -4 + 4 = 0.
Hyogo Entrance Math practice questions
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Calculate the value of 4 - 2 × (-3) + (-12) ÷ 3.
- -6
- 14
- -2
- 6 Correct answer.
Explanation
Multiplication and division come before addition and subtraction: -2 × (-3) = +6 and (-12) ÷ 3 = -4. The expression becomes 4 + 6 - 4 = 6.
Calculate 2/3 + 3/4 - 1/6.
- 3/4
- 13/12
- 7/12
- 5/4 Correct answer.
Explanation
Use the common denominator 12: 2/3 = 8/12, 3/4 = 9/12, and 1/6 = 2/12. Then 8/12 + 9/12 - 2/12 = 15/12 = 5/4.
Evaluate the expression 3x - 2y when x = 5 and y = -3.
- -9
- 9
- -21
- 21 Correct answer.
Explanation
Substitute the values: 3(5) - 2(-3) = 15 + 6 = 21. The key step is that -2y with y = -3 gives -2 × (-3) = +6.
Expand (x + 4)(x + 7).
- x² + 11x + 28 Correct answer.
- x² + 11x + 11
- x² + 28x + 11
- x² + 3x + 28
Explanation
Using (x + a)(x + b) = x² + (a + b)x + ab with a = 4 and b = 7: the coefficient of x is 4 + 7 = 11 and the constant term is 4 × 7 = 28. The result is x² + 11x + 28.
Factor the expression x² - 16.
- (x - 4)(x + 4) Correct answer.
- (x + 4)²
- (x - 16)(x + 1)
- (x - 4)²
Explanation
This is a difference of squares: x² - 16 = x² - 4². Applying a² - b² = (a + b)(a - b) gives (x + 4)(x - 4).
Simplify the expression 4(2a - b) - 3(a - 2b).
- 5a
- 5a - 10b
- 5a + 2b Correct answer.
- 5a - 2b
Explanation
Distribute each factor: 4(2a - b) = 8a - 4b and -3(a - 2b) = -3a + 6b. Combining like terms gives (8a - 3a) + (-4b + 6b) = 5a + 2b.
Calculate √20 + √45 - √5.
- 4√5 Correct answer.
- 3√5
- 5√5
- √60
Explanation
Simplify each radical: √20 = 2√5, √45 = 3√5, and √5 stays as is. Then 2√5 + 3√5 - √5 = 4√5.
Rewrite 8/√2 with a rationalized denominator.
- 4
- 4√2 Correct answer.
- 2√2
- 8√2
Explanation
Multiply numerator and denominator by √2: (8 × √2)/(√2 × √2) = 8√2/2. Dividing 8 by 2 gives 4√2.
Expand and simplify (x + 5)² - (x - 5)².
- 10x
- 50
- 0
- 20x Correct answer.
Explanation
Expand each square: (x + 5)² = x² + 10x + 25 and (x - 5)² = x² - 10x + 25. Subtracting gives (x² + 10x + 25) - (x² - 10x + 25) = 20x, since the x² and constant terms cancel.
Factor the expression x² + 9x + 20.
- (x + 2)(x + 10)
- (x - 4)(x - 5)
- (x + 4)(x + 5) Correct answer.
- (x + 1)(x + 20)
Explanation
We need two numbers whose sum is 9 and whose product is 20. Those numbers are 4 and 5, so x² + 9x + 20 = (x + 4)(x + 5).
Factor the expression 5x² - 20 completely.
- 5(x - 2)²
- (5x - 4)(x + 5)
- (5x + 2)(5x - 2)
- 5(x + 2)(x - 2) Correct answer.
Explanation
First factor out the common factor 5: 5x² - 20 = 5(x² - 4). Then x² - 4 is a difference of squares, giving 5(x + 2)(x - 2).
Evaluate x² - 2xy + y² when x = 15 and y = 7.
- 484
- -56
- 64 Correct answer.
- 176
Explanation
The expression factors as (x - y)², so compute (15 - 7)² = 8² = 64. Recognizing the perfect-square form avoids the messy arithmetic of 225 - 210 + 49.
Simplify (3/4)a²b × (-8ab³) ÷ (2a²b²).
- -3ab
- 3ab²
- -12ab²
- -3ab² Correct answer.
Explanation
Multiply the coefficients: (3/4) × (-8) ÷ 2 = -6 ÷ 2 = -3. For the variables, a² × a ÷ a² = a and b × b³ ÷ b² = b², so the result is -3ab².
Calculate (√7 + √2)(√7 - √2) + √24 ÷ √6.
- 5 + 2√6
- 5
- 9
- 7 Correct answer.
Explanation
The product is a difference of squares: (√7)² - (√2)² = 7 - 2 = 5. The division gives √(24/6) = √4 = 2, so the total is 5 + 2 = 7.
Given x + y = 9 and xy = 14, find the value of x² + y².
- 53 Correct answer.
- 81
- 67
- 109
Explanation
Use the identity x² + y² = (x + y)² - 2xy. Substituting gives 9² - 2(14) = 81 - 28 = 53.
Factor the expression x² - 7x - 30.
- (x - 5)(x + 6)
- (x - 15)(x + 2)
- (x + 10)(x - 3)
- (x - 10)(x + 3) Correct answer.
Explanation
We need two numbers with product -30 and sum -7. Those numbers are -10 and +3, so x² - 7x - 30 = (x - 10)(x + 3).
Calculate (-1/3) + (3/4).
- 2/7
- -13/12
- 5/12 Correct answer.
- -5/12
Explanation
Convert to the common denominator 12: -1/3 = -4/12 and 3/4 = 9/12. Then -4/12 + 9/12 = 5/12.
Simplify (x - 3)/4 + (x + 2)/3.
- (7x - 5)/12
- (2x - 1)/7
- (7x - 1)/7
- (7x - 1)/12 Correct answer.
Explanation
Use the common denominator 12: 3(x - 3)/12 + 4(x + 2)/12 = (3x - 9 + 4x + 8)/12. Combining like terms in the numerator gives (7x - 1)/12.
Calculate √3 × √15 ÷ √5.
- 3√3
- √3
- 9
- 3 Correct answer.
Explanation
Combine under one root: √(3 × 15 ÷ 5) = √9 = 3. Equivalently, √3 × √15 = √45 = 3√5, and 3√5 ÷ √5 = 3.
Solve the equation 4x - 5 = 11.
- x = 6
- x = 3/2
- x = 4 Correct answer.
- x = 16
Explanation
Add 5 to both sides: 4x = 16. Then divide both sides by 4: x = 4.
Solve the equation -3x = 18.
- x = -15
- x = 6
- x = 54
- x = -6 Correct answer.
Explanation
Divide both sides by -3: x = 18 ÷ (-3) = -6. Dividing a positive number by a negative number gives a negative result.
Solve the simultaneous equations: x + y = 12 and x - y = 6.
- x = 3, y = 9
- x = 9, y = 3 Correct answer.
- x = 18, y = 6
- x = 6, y = 6
Explanation
Add the two equations to eliminate y: 2x = 18, so x = 9. Substitute into the first equation: 9 + y = 12, giving y = 3.
Solve the equation x² = 49.
- x = 7 only
- x = 49
- x = 7 and x = -7 Correct answer.
- x = 24.5
Explanation
Taking the square root of both sides gives x = ±√49 = ±7. Both 7 and -7 satisfy the equation because (-7)² = 49 as well.
Solve the equation x² - 7x + 12 = 0.
- x = 3 and x = 4 Correct answer.
- x = -1 and x = -12
- x = 1 and x = 12
- x = -3 and x = -4
Explanation
Factor the left side: x² - 7x + 12 = (x - 3)(x - 4). Setting each factor to zero gives x = 3 or x = 4.
Solve the equation 4(x - 3) = 2x + 6.
- x = 18
- x = 3
- x = 9 Correct answer.
- x = -9
Explanation
Expand the left side: 4x - 12 = 2x + 6. Subtract 2x from both sides: 2x - 12 = 6, then add 12: 2x = 18, so x = 9.
Solve the simultaneous equations: 3x + 2y = 16 and x - y = 2.
- x = 6, y = 2
- x = 4, y = 2 Correct answer.
- x = 5, y = 3
- x = 2, y = 4
Explanation
From the second equation, x = y + 2. Substitute into the first: 3(y + 2) + 2y = 16, so 5y + 6 = 16 and y = 2. Then x = 2 + 2 = 4.
Solve the equation x² + 8x + 15 = 0.
- x = 1 and x = 15
- x = -3 and x = -5 Correct answer.
- x = -1 and x = -15
- x = 3 and x = 5
Explanation
Factor: x² + 8x + 15 = (x + 3)(x + 5). Setting each factor to zero gives x = -3 or x = -5.
Solve the equation 3x² - 10x - 8 = 0.
- x = -2/3 and x = -4
- x = -2/3 and x = 4 Correct answer.
- x = 2/3 and x = -4
- x = 2/3 and x = 4
Explanation
The equation factors as (3x + 2)(x - 4) = 0, giving x = -2/3 or x = 4. Check with the quadratic formula: x = (10 ± √(100 + 96))/6 = (10 ± 14)/6, which gives 4 and -2/3.
Solve the inequality 4x - 3 < 9.
- x < 3/4
- x < 3 Correct answer.
- x < 12
- x > 3
Explanation
Add 3 to both sides: 4x < 12. Dividing by 4 (a positive number, so the direction stays) gives x < 3.
Solve the inequality 3 - 2x ≤ 9.
- x ≤ 3
- x ≥ -10/3
- x ≤ -3
- x ≥ -3 Correct answer.
Explanation
Subtract 3 from both sides: -2x ≤ 6. Dividing by -2 reverses the inequality sign, giving x ≥ -3.
A shirt is sold at a 25% discount for 1500 yen. What was the original price?
- 1125 yen
- 2000 yen Correct answer.
- 1875 yen
- 1750 yen
Explanation
Let the original price be x yen. After a 25% discount, 0.75x = 1500, so x = 1500 ÷ 0.75 = 2000 yen.
A walker goes from town A to town B, 30 km apart, at 6 km/h, and returns along the same road at 5 km/h. How many hours does the round trip take?
- 2.5 hours
- 11 hours Correct answer.
- 5.5 hours
- 10 hours
Explanation
The outward trip takes 30 ÷ 6 = 5 hours and the return takes 30 ÷ 5 = 6 hours. The total is 5 + 6 = 11 hours.
The sum of the digits of a two-digit number is 13. When the digits are reversed, the new number is 45 greater than the original. Find the original number.
- 49 Correct answer.
- 85
- 94
- 58
Explanation
Let the tens digit be a and the units digit be b. Then a + b = 13 and (10b + a) - (10a + b) = 45, which simplifies to 9(b - a) = 45, so b - a = 5. Solving the system gives a = 4 and b = 9, so the number is 49.
How many grams of a 10% salt solution must be mixed with a 4% salt solution to make 400 g of a 7% salt solution?
- 250 g
- 100 g
- 150 g
- 200 g Correct answer.
Explanation
Let x be the grams of 10% solution; then (400 - x) is the grams of 4% solution. The salt equation is 0.10x + 0.04(400 - x) = 0.07 × 400 = 28, which gives 0.06x + 16 = 28, so x = 200 g.
Solve the equation x² - 8x + 3 = 0.
- x = 4 ± √52
- x = 4 ± 2√13
- x = 4 ± √13 Correct answer.
- x = 8 ± √13
Explanation
Using the quadratic formula: x = (8 ± √(64 - 12))/2 = (8 ± √52)/2. Since √52 = 2√13, this simplifies to (8 ± 2√13)/2 = 4 ± √13.
Solve the equation (x + 2)² = 3x + 4.
- x = 0 and x = -1 Correct answer.
- x = 2 ± √3
- x = 2 and x = -1
- x = 1 and x = -1
Explanation
Expand the left side: x² + 4x + 4 = 3x + 4. Move everything to one side: x² + x = 0, which factors as x(x + 1) = 0. The solutions are x = 0 and x = -1.
A student has 2000 yen. After buying a 480-yen item, she wants to buy as many 120-yen notebooks as possible. What is the maximum number of notebooks she can buy?
- 11 notebooks
- 12 notebooks Correct answer.
- 10 notebooks
- 13 notebooks
Explanation
Let x be the number of notebooks. Then 480 + 120x ≤ 2000, so 120x ≤ 1520 and x ≤ 12.66.... Since x must be a whole number, the maximum is 12 notebooks.
For the direct proportion y = 4x, find the value of y when x = 6.
- 24 Correct answer.
- 1296
- 2/3
- 10
Explanation
Substitute x = 6 into the equation: y = 4 × 6 = 24. In a direct proportion, y is simply the constant multiplied by x.
y is inversely proportional to x, and y = 8 when x = 3. Find the equation relating x and y.
- y = 8/x
- y = 11/x
- y = 3x
- y = 24/x Correct answer.
Explanation
For inverse proportion y = a/x, the constant is a = xy = 3 × 8 = 24. So the equation is y = 24/x.
What is the slope of the line y = -3x + 7?
- -3 Correct answer.
- 7
- -7
- 3
Explanation
In the form y = ax + b, the slope is a, the coefficient of x. Here a = -3, so the slope is -3; the 7 is the y-intercept.
For the function y = 3x², find the value of y when x = 4.
- 24
- 12
- 144
- 48 Correct answer.
Explanation
Substitute x = 4: y = 3 × 4² = 3 × 16 = 48. Note that the square applies to x before multiplying by 3.
The point (3, 15) lies on the graph of y = ax. Find the value of a.
- 5 Correct answer.
- 1/5
- 12
- 45
Explanation
Substitute x = 3 and y = 15: 15 = 3a, so a = 5. The constant of proportionality is always y ÷ x.
A line has slope -2 and passes through the point (0, 4). Find the equation of the line.
- y = -2x + 4 Correct answer.
- y = 4x - 2
- y = -2x - 4
- y = 2x + 4
Explanation
A line with slope -2 has the form y = -2x + b. Since the point (0, 4) is on the y-axis, the y-intercept is 4, so the equation is y = -2x + 4.
Find the equation of the line passing through the points (2, 5) and (4, 11).
- y = 3x - 1 Correct answer.
- y = 2x + 1
- y = 3x
- y = 3x + 1
Explanation
The slope is (11 - 5)/(4 - 2) = 6/2 = 3. Using y = 3x + b with the point (2, 5): 5 = 3(2) + b, so b = -1. The equation is y = 3x - 1.
Find the coordinates of the intersection point of the lines y = 3x - 2 and y = -x + 6.
- (2, 4) Correct answer.
- (2, 5)
- (4, 2)
- (1, 3)
Explanation
Set the expressions equal: 3x - 2 = -x + 6, so 4x = 8 and x = 2. Substituting into y = 3x - 2 gives y = 4, so the intersection is (2, 4).
y is inversely proportional to x, and y = 6 when x = 4. Find the value of y when x = 8.
- 3 Correct answer.
- 12
- 8
- 2
Explanation
The constant of proportionality is a = xy = 4 × 6 = 24, so y = 24/x. When x = 8, y = 24 ÷ 8 = 3.
For the function y = 2x², find the rate of change as x increases from 2 to 5.
- 42/5
- 10
- 14 Correct answer.
- 7
Explanation
The rate of change is (change in y) ÷ (change in x) = (2×25 - 2×4)/(5 - 2) = (50 - 8)/3 = 42/3 = 14. For y = ax² this equals a(sum of the x-values) = 2(2 + 5) = 14.
For the function y = -x² with domain -3 ≤ x ≤ 2, find the range of y.
- 0 ≤ y ≤ 9
- -9 ≤ y ≤ -4
- -9 ≤ y ≤ 0 Correct answer.
- -4 ≤ y ≤ 0
Explanation
Since the domain includes x = 0, the maximum value is y = 0 at the vertex. The minimum occurs at the endpoint farthest from 0: at x = -3, y = -(-3)² = -9. So the range is -9 ≤ y ≤ 0.
The line y = 2x - 4 meets the x-axis and y-axis at two points. Find the area of the triangle these two points form with the origin.
- 2
- 1
- 4 Correct answer.
- 8
Explanation
The y-intercept is (0, -4) and the x-intercept is found from 0 = 2x - 4, giving (2, 0). The triangle has legs of length 4 along the y-axis and 2 along the x-axis, so its area is (1/2) × 2 × 4 = 4.