Calculate the value of (-6) + (+4) - (-3).
- 1 Correct answer.
- -5
- 13
- -7
Explanation
Subtracting a negative number becomes addition, so -(-3) = +3. The expression becomes -6 + 4 + 3. Computing left to right: -6 + 4 = -2, then -2 + 3 = 1.
Hokkaido Entrance Math practice questions
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Calculate the value of 5 - 3 × (-2) + (-8) ÷ 2.
- 15
- -5
- -8
- 7 Correct answer.
Explanation
Multiplication and division come before addition and subtraction: -3 × (-2) = +6 and (-8) ÷ 2 = -4. The expression becomes 5 + 6 - 4 = 7.
Calculate 3/4 + 5/6 - 1/2.
- 7/6
- 5/12
- 19/12
- 13/12 Correct answer.
Explanation
Use the common denominator 12: 3/4 = 9/12, 5/6 = 10/12, and 1/2 = 6/12. Then 9/12 + 10/12 - 6/12 = 13/12, which is already in simplest form.
Evaluate the expression 2x - 3y when x = 4 and y = -2.
- -2
- -14
- 2
- 14 Correct answer.
Explanation
Substitute the values: 2(4) - 3(-2) = 8 + 6 = 14. The key step is that -3y with y = -2 gives -3 × (-2) = +6.
Simplify the expression 3(2a - b) - 2(a - 3b).
- 4a + 5b
- 4a - 9b
- 4a
- 4a + 3b Correct answer.
Explanation
Distribute each factor: 3(2a - b) = 6a - 3b and -2(a - 3b) = -2a + 6b. Combining like terms gives (6a - 2a) + (-3b + 6b) = 4a + 3b.
Expand (x + 3)(x + 5).
- x² + 8x + 8
- x² + 15x + 8
- x² + 2x + 15
- x² + 8x + 15 Correct answer.
Explanation
Using (x + a)(x + b) = x² + (a + b)x + ab with a = 3 and b = 5: the coefficient of x is 3 + 5 = 8 and the constant term is 3 × 5 = 15. The result is x² + 8x + 15.
Factor the expression x² - 9.
- (x - 3)²
- (x + 3)²
- (x - 9)(x + 1)
- (x - 3)(x + 3) Correct answer.
Explanation
This is a difference of squares: x² - 9 = x² - 3². Applying a² - b² = (a + b)(a - b) gives (x + 3)(x - 3).
Calculate √18 + √8 - √2.
- 4√2 Correct answer.
- 4
- 5√2
- 3√2
Explanation
Simplify each radical: √18 = 3√2, √8 = 2√2, and √2 stays as is. Then 3√2 + 2√2 - √2 = 4√2.
Rewrite 6/√3 with a rationalized denominator.
- 6√3
- 2√3 Correct answer.
- √3/2
- 3√3
Explanation
Multiply numerator and denominator by √3: (6 × √3)/(√3 × √3) = 6√3/3. Dividing 6 by 3 gives 2√3.
Expand and simplify (x + 4)² - (x - 4)².
- 8x
- 0
- 16x Correct answer.
- 32
Explanation
Expand each square: (x + 4)² = x² + 8x + 16 and (x - 4)² = x² - 8x + 16. Subtracting gives (x² + 8x + 16) - (x² - 8x + 16) = 16x, since the x² and constant terms cancel.
Factor the expression x² + 7x + 12.
- (x + 1)(x + 12)
- (x + 3)(x + 4) Correct answer.
- (x + 2)(x + 6)
- (x - 3)(x - 4)
Explanation
We need two numbers whose sum is 7 and whose product is 12. Those numbers are 3 and 4, so x² + 7x + 12 = (x + 3)(x + 4).
Factor the expression 3x² - 12 completely.
- (3x - 4)(x + 3)
- 3(x - 2)²
- 3(x + 2)(x - 2) Correct answer.
- (3x + 2)(3x - 2)
Explanation
First factor out the common factor 3: 3x² - 12 = 3(x² - 4). Then x² - 4 is a difference of squares, giving 3(x + 2)(x - 2).
Evaluate x² - 2xy + y² when x = 13 and y = 8.
- 25 Correct answer.
- -39
- 105
- 441
Explanation
The expression factors as (x - y)², so compute (13 - 8)² = 5² = 25. Recognizing the perfect-square form avoids the messy arithmetic of 169 - 208 + 64.
Simplify (2/3)ab² × (-6a²b) ÷ (4ab³).
- -16a²
- a²
- -a²b³
- -a² Correct answer.
Explanation
Multiply the coefficients: (2/3) × (-6) ÷ 4 = -4 ÷ 4 = -1. For the variables, a × a² ÷ a = a² and b² × b ÷ b³ = b⁰ = 1, so the result is -a².
Calculate (√5 + √3)(√5 - √3) + √18 ÷ √2.
- 2 + 3√2
- 8
- 3
- 5 Correct answer.
Explanation
The product is a difference of squares: (√5)² - (√3)² = 5 - 3 = 2. The division gives √(18/2) = √9 = 3, so the total is 2 + 3 = 5.
Given x + y = 7 and xy = 10, find the value of x² + y².
- 29 Correct answer.
- 49
- 69
- 39
Explanation
Use the identity x² + y² = (x + y)² - 2xy. Substituting gives 7² - 2(10) = 49 - 20 = 29.
Factor the expression x² - 5x - 36.
- (x + 9)(x - 4)
- (x - 9)(x + 4) Correct answer.
- (x - 12)(x + 3)
- (x - 6)(x + 6)
Explanation
We need two numbers with product -36 and sum -5. Those numbers are -9 and +4, so x² - 5x - 36 = (x - 9)(x + 4).
Calculate (-1/2) + (2/3).
- 1/5
- 1/6 Correct answer.
- -1/6
- -7/6
Explanation
Convert to the common denominator 6: -1/2 = -3/6 and 2/3 = 4/6. Then -3/6 + 4/6 = 1/6.
Simplify (x - 2)/3 + (x + 1)/4.
- (2x - 1)/7
- (7x - 5)/7
- (7x - 5)/12 Correct answer.
- (7x - 11)/12
Explanation
Use the common denominator 12: 4(x - 2)/12 + 3(x + 1)/12 = (4x - 8 + 3x + 3)/12. Combining like terms in the numerator gives (7x - 5)/12.
Calculate √2 × √10 ÷ √5.
- √10
- 4
- 2√5
- 2 Correct answer.
Explanation
Combine under one root: √(2 × 10 ÷ 5) = √4 = 2. Equivalently, √2 × √10 = √20 = 2√5, and 2√5 ÷ √5 = 2.
Solve the equation 3x - 7 = 8.
- x = 5 Correct answer.
- x = 1/3
- x = 15
- x = 3
Explanation
Add 7 to both sides: 3x = 15. Then divide both sides by 3: x = 5.
Solve the equation -2x = 14.
- x = 7
- x = -7 Correct answer.
- x = -12
- x = 28
Explanation
Divide both sides by -2: x = 14 ÷ (-2) = -7. Dividing a positive number by a negative number gives a negative result.
Solve the simultaneous equations: x + y = 10 and x - y = 4.
- x = 3, y = 7
- x = 7, y = 3 Correct answer.
- x = 14, y = 6
- x = 6, y = 4
Explanation
Add the two equations to eliminate y: 2x = 14, so x = 7. Substitute into the first equation: 7 + y = 10, giving y = 3.
Solve the equation x² = 25.
- x = 12.5
- x = 5 only
- x = 25
- x = 5 and x = -5 Correct answer.
Explanation
Taking the square root of both sides gives x = ±√25 = ±5. Both 5 and -5 satisfy the equation because (-5)² = 25 as well.
Solve the equation x² - 5x + 6 = 0.
- x = -1 and x = -6
- x = 2 and x = 3 Correct answer.
- x = 1 and x = 6
- x = -2 and x = -3
Explanation
Factor the left side: x² - 5x + 6 = (x - 2)(x - 3). Setting each factor to zero gives x = 2 or x = 3.
Solve the equation 5(x - 2) = 3x + 4.
- x = 7 Correct answer.
- x = -3
- x = 14
- x = 3
Explanation
Expand the left side: 5x - 10 = 3x + 4. Subtract 3x from both sides: 2x - 10 = 4, then add 10: 2x = 14, so x = 7.
Solve the simultaneous equations: 2x + 3y = 12 and x - y = 1.
- x = 4, y = 2
- x = 5, y = 4
- x = 3, y = 2 Correct answer.
- x = 2, y = 3
Explanation
From the second equation, x = y + 1. Substitute into the first: 2(y + 1) + 3y = 12, so 5y + 2 = 12 and y = 2. Then x = 2 + 1 = 3.
Solve the equation x² + 7x + 10 = 0.
- x = 2 and x = 5
- x = -2 and x = -5 Correct answer.
- x = -1 and x = -10
- x = 1 and x = 10
Explanation
Factor: x² + 7x + 10 = (x + 2)(x + 5). Setting each factor to zero gives x = -2 or x = -5.
Solve the equation 2x² - 5x - 3 = 0.
- x = -3 and x = -1/2
- x = 3 and x = 1/2
- x = -3 and x = 1/2
- x = 3 and x = -1/2 Correct answer.
Explanation
The equation factors as (2x + 1)(x - 3) = 0, giving x = -1/2 or x = 3. Check with the quadratic formula: x = (5 ± √(25 + 24))/4 = (5 ± 7)/4, which gives 3 and -1/2.
Solve the inequality 3x - 5 > 7.
- x > 12
- x > 2/3
- x > 4 Correct answer.
- x < 4
Explanation
Add 5 to both sides: 3x > 12. Dividing by 3 (a positive number, so the direction stays) gives x > 4.
Solve the inequality 2 - 3x ≥ 8.
- x ≥ -2
- x ≤ 2
- x ≥ -10/3
- x ≤ -2 Correct answer.
Explanation
Subtract 2 from both sides: -3x ≥ 6. Dividing by -3 reverses the inequality sign, giving x ≤ -2.
A book is sold at a 20% discount for 960 yen. What was the original price?
- 1152 yen
- 1000 yen
- 768 yen
- 1200 yen Correct answer.
Explanation
Let the original price be x yen. After a 20% discount, 0.8x = 960, so x = 960 ÷ 0.8 = 1200 yen.
A cyclist rides from town A to town B, 24 km apart, at 12 km/h, and returns along the same road at 8 km/h. How many hours does the round trip take?
- 6 hours
- 2.4 hours
- 4.8 hours
- 5 hours Correct answer.
Explanation
The outward trip takes 24 ÷ 12 = 2 hours and the return takes 24 ÷ 8 = 3 hours. The total is 2 + 3 = 5 hours.
The sum of the digits of a two-digit number is 11. When the digits are reversed, the new number is 27 greater than the original. Find the original number.
- 74
- 56
- 38
- 47 Correct answer.
Explanation
Let the tens digit be a and the units digit be b. Then a + b = 11 and (10b + a) - (10a + b) = 27, which simplifies to 9(b - a) = 27, so b - a = 3. Solving the system gives a = 4 and b = 7, so the number is 47.
How many grams of an 8% salt solution must be mixed with a 3% salt solution to make 500 g of a 5% salt solution?
- 250 g
- 300 g
- 100 g
- 200 g Correct answer.
Explanation
Let x be the grams of 8% solution; then (500 - x) is the grams of 3% solution. The salt equation is 0.08x + 0.03(500 - x) = 0.05 × 500 = 25, which gives 0.05x + 15 = 25, so x = 200 g.
Solve the equation x² - 6x + 4 = 0.
- x = 3 ± 2√5
- x = 3 ± √5 Correct answer.
- x = 3 ± √20
- x = 6 ± √5
Explanation
Using the quadratic formula: x = (6 ± √(36 - 16))/2 = (6 ± √20)/2. Since √20 = 2√5, this simplifies to (6 ± 2√5)/2 = 3 ± √5.
Solve the equation (x - 1)² = 2x - 1.
- x = 2 ± √2 Correct answer.
- x = 1 ± √2
- x = 4 ± 2√2
- x = 2 ± 2√2
Explanation
Expand the left side: x² - 2x + 1 = 2x - 1. Move everything to one side: x² - 4x + 2 = 0. The quadratic formula gives x = (4 ± √(16 - 8))/2 = (4 ± 2√2)/2 = 2 ± √2.
A student has 1000 yen. After buying a 250-yen lunch, she wants to buy as many 90-yen notebooks as possible. What is the maximum number of notebooks she can buy?
- 10 notebooks
- 7 notebooks
- 9 notebooks
- 8 notebooks Correct answer.
Explanation
Let x be the number of notebooks. Then 250 + 90x ≤ 1000, so 90x ≤ 750 and x ≤ 8.33.... Since x must be a whole number, the maximum is 8 notebooks.
For the direct proportion y = 3x, find the value of y when x = 4.
- 7
- 4/3
- 12 Correct answer.
- 81
Explanation
Substitute x = 4 into the equation: y = 3 × 4 = 12. In a direct proportion, y is simply the constant multiplied by x.
y is inversely proportional to x, and y = 6 when x = 2. Find the equation relating x and y.
- y = 12/x Correct answer.
- y = 6/x
- y = 3x
- y = 8/x
Explanation
For inverse proportion y = a/x, the constant is a = xy = 2 × 6 = 12. So the equation is y = 12/x.
What is the slope of the line y = -2x + 5?
- -2 Correct answer.
- 2
- -5
- 5
Explanation
In the form y = ax + b, the slope is a, the coefficient of x. Here a = -2, so the slope is -2; the 5 is the y-intercept.
For the function y = 2x², find the value of y when x = 3.
- 18 Correct answer.
- 12
- 6
- 36
Explanation
Substitute x = 3: y = 2 × 3² = 2 × 9 = 18. Note that the square applies to x before multiplying by 2.
The point (2, 6) lies on the graph of y = ax. Find the value of a.
- 12
- 4
- 3 Correct answer.
- 1/3
Explanation
Substitute x = 2 and y = 6: 6 = 2a, so a = 3. The constant of proportionality is always y ÷ x.
A line has slope 2 and passes through the point (0, -3). Find the equation of the line.
- y = -3x + 2
- y = 2x - 3 Correct answer.
- y = 3x - 2
- y = 2x + 3
Explanation
A line with slope 2 has the form y = 2x + b. Since the point (0, -3) is on the y-axis, the y-intercept is -3, so the equation is y = 2x - 3.
Find the equation of the line passing through the points (1, 3) and (3, 9).
- y = 3x Correct answer.
- y = 3x + 1
- y = 2x + 1
- y = 6x - 3
Explanation
The slope is (9 - 3)/(3 - 1) = 6/2 = 3. Using y = 3x + b with the point (1, 3): 3 = 3(1) + b, so b = 0. The equation is y = 3x.
Find the coordinates of the intersection point of the lines y = 2x + 1 and y = -x + 7.
- (3, 4)
- (2, 5) Correct answer.
- (2, 3)
- (5, 2)
Explanation
Set the expressions equal: 2x + 1 = -x + 7, so 3x = 6 and x = 2. Substituting into y = 2x + 1 gives y = 5, so the intersection is (2, 5).
y is inversely proportional to x, and y = 4 when x = 6. Find the value of y when x = 8.
- 16/3
- 2
- 12
- 3 Correct answer.
Explanation
The constant of proportionality is a = xy = 6 × 4 = 24, so y = 24/x. When x = 8, y = 24 ÷ 8 = 3.
For the function y = x², find the rate of change (average rate) as x increases from 1 to 4.
- 4
- 3
- 15/4
- 5 Correct answer.
Explanation
The rate of change is (change in y) ÷ (change in x) = (16 - 1)/(4 - 1) = 15/3 = 5. For y = ax² this equals a(sum of the x-values) = 1 + 4 = 5.
For the function y = (1/2)x² with domain -2 ≤ x ≤ 4, find the range of y.
- 2 ≤ y ≤ 8
- -2 ≤ y ≤ 8
- 0 ≤ y ≤ 8 Correct answer.
- 0 ≤ y ≤ 2
Explanation
Since the domain includes x = 0, the minimum value is y = 0 at the vertex. The maximum occurs at the endpoint farthest from 0: at x = 4, y = (1/2)(16) = 8. So the range is 0 ≤ y ≤ 8.
The line y = x + 2 meets the x-axis and y-axis at two points. Find the area of the triangle these two points form with the origin.
- 1
- 2 Correct answer.
- 4
- 8
Explanation
The y-intercept is (0, 2) and the x-intercept is found from 0 = x + 2, giving (-2, 0). The triangle has legs of length 2 along each axis, so its area is (1/2) × 2 × 2 = 2.