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Hiroshima Entrance Math study materials
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Hiroshima Entrance Math practice questions

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Question 1Numbers and Algebraic Expressions

Calculate the value of (-7) + (+3) - (-5).

  1. 15
  2. 1 Correct answer.
  3. -9
  4. -5

Explanation

Subtracting a negative number becomes addition, so -(-5) = +5. The expression is -7 + 3 + 5. Computing left to right: -7 + 3 = -4, then -4 + 5 = 1.

Question 2Numbers and Algebraic Expressions

Calculate the value of (-3)² - 4 × (-2) + 10 ÷ 5.

  1. 19 Correct answer.
  2. 3
  3. 1
  4. 15

Explanation

Multiplication, division, and powers come before addition and subtraction. (-3)² = 9, -4 × (-2) = +8, and 10 ÷ 5 = 2, so the expression is 9 + 8 + 2 = 19.

Question 3Numbers and Algebraic Expressions

Calculate 2/3 - 5/6 + 3/4.

  1. 9/4
  2. 0
  3. 7/12 Correct answer.
  4. -1/6

Explanation

Use the common denominator 12: 2/3 = 8/12, 5/6 = 10/12, and 3/4 = 9/12. Then 8/12 - 10/12 + 9/12 = 7/12, which is already in simplest form.

Question 4Numbers and Algebraic Expressions

Evaluate the expression 3a - 2b when a = -2 and b = 4.

  1. 2
  2. -2
  3. 14
  4. -14 Correct answer.

Explanation

Substitute the values: 3(-2) - 2(4) = -6 - 8 = -14. Both products are negative, so the result is -14.

Question 5Numbers and Algebraic Expressions

Calculate (-3/4) ÷ (9/8).

  1. -2/3 Correct answer.
  2. -27/32
  3. 2/3
  4. -3/2

Explanation

Dividing by a fraction means multiplying by its reciprocal: (-3/4) ÷ (9/8) = (-3/4) × (8/9) = -24/36. Cancelling the common factor 12 simplifies this to -2/3.

Question 6Numbers and Algebraic Expressions

Simplify √48 + √27 - √12.

  1. 7√3
  2. 9√3
  3. 5√3 Correct answer.
  4. 3√3

Explanation

Simplify each radical: √48 = 4√3, √27 = 3√3, and √12 = 2√3. Then 4√3 + 3√3 - 2√3 = (4 + 3 - 2)√3 = 5√3.

Question 7Numbers and Algebraic Expressions

Expand (x - 4)(x + 7).

  1. x² + 11x - 28
  2. x² - 3x - 28
  3. x² + 3x - 28 Correct answer.
  4. x² + 3x + 28

Explanation

Using the expansion formula, (x - 4)(x + 7) = x² + (-4 + 7)x + (-4) × 7 = x² + 3x - 28. The middle coefficient is the signed sum -4 + 7 = 3, and the constant is the product -28.

Question 8Numbers and Algebraic Expressions

Factorize x² + 7x + 12.

  1. (x - 3)(x - 4)
  2. (x + 1)(x + 12)
  3. (x + 3)(x + 4) Correct answer.
  4. (x + 2)(x + 6)

Explanation

Look for two numbers whose product is 12 and whose sum is 7. Those numbers are 3 and 4, so x² + 7x + 12 = (x + 3)(x + 4).

Question 9Numbers and Algebraic Expressions

Simplify 2(3x - 2) - (4x - 7).

  1. 2x - 3
  2. 2x + 3 Correct answer.
  3. 2x - 11
  4. -x + 5

Explanation

Distribute first: 2(3x - 2) = 6x - 4, and -(4x - 7) = -4x + 7. Combining gives 6x - 4 - 4x + 7 = 2x + 3.

Question 10Numbers and Algebraic Expressions

Tiles are arranged in a cross pattern: the 1st figure uses 5 tiles, and each new figure adds 4 tiles, so the 2nd uses 9 and the 3rd uses 13. How many tiles does the 25th figure use?

  1. 101 Correct answer.
  2. 125
  3. 105
  4. 100

Explanation

The first figure uses 5 tiles and each later figure adds 4, so the nth figure uses 5 + 4(n - 1) = 4n + 1 tiles. Substituting n = 25 gives 4(25) + 1 = 101 tiles.

Question 11Numbers and Algebraic Expressions

Evaluate x² - 3xy when x = 3 and y = -2.

  1. 18
  2. -9
  3. 27 Correct answer.
  4. 24

Explanation

Substitute the values: x² = 9 and -3xy = -3(3)(-2) = +18. So the value is 9 + 18 = 27.

Question 12Numbers and Algebraic Expressions

Simplify (3/2)√8 - √2.

  1. 4√2
  2. 3√2
  3. 2√2 Correct answer.
  4. √2

Explanation

Since √8 = 2√2, the first term is (3/2) × 2√2 = 3√2. Then 3√2 - √2 = 2√2.

Question 13Numbers and Algebraic Expressions

Find the smallest positive integer n such that √(54n) is an integer.

  1. 3
  2. 4
  3. 2
  4. 6 Correct answer.

Explanation

Factorize 54 = 2 × 3³. For 54n to be a perfect square, every prime must appear to an even power, so n must supply one more 2 and one more 3: n = 2 × 3 = 6. Indeed 54 × 6 = 324 = 18².

Question 14Numbers and Algebraic Expressions

Calculate (-2)³ × 5 + 18 ÷ (-3).

  1. 34
  2. -34
  3. -46 Correct answer.
  4. -36

Explanation

Compute each piece: (-2)³ = -8, so (-2)³ × 5 = -40, and 18 ÷ (-3) = -6. Then -40 + (-6) = -46.

Question 15Numbers and Algebraic Expressions

Expand (3x - 4)².

  1. 9x² - 16
  2. 3x² - 24x + 16
  3. 9x² - 24x + 16 Correct answer.
  4. 9x² + 16

Explanation

Use the formula (a - b)² = a² - 2ab + b² with a = 3x and b = 4. This gives (3x)² - 2(3x)(4) + 4² = 9x² - 24x + 16.

Question 16Numbers and Algebraic Expressions

Factorize x² - 9x + 20.

  1. (x + 4)(x - 5)
  2. (x - 4)(x - 5) Correct answer.
  3. (x - 2)(x - 10)
  4. (x + 4)(x + 5)

Explanation

Look for two numbers whose product is +20 and whose sum is -9. Those numbers are -4 and -5, so x² - 9x + 20 = (x - 4)(x - 5).

Question 17Numbers and Algebraic Expressions

Given x - y = 3 and xy = 5, find the value of x² + y².

  1. 9
  2. 19 Correct answer.
  3. 24
  4. -1

Explanation

Expanding (x - y)² gives x² - 2xy + y², so x² + y² = (x - y)² + 2xy. Substituting gives 3² + 2(5) = 9 + 10 = 19. Note that when the given expression is a difference, the 2xy term is ADDED back.

Question 18Numbers and Algebraic Expressions

When a = -3, find the value of -a² + 4a.

  1. 21
  2. -21 Correct answer.
  3. 3
  4. -3

Explanation

Substitute a = -3: -a² = -(-3)² = -9 and 4a = -12. Then -9 + (-12) = -21.

Question 19Numbers and Algebraic Expressions

Find the 30th term of the arithmetic sequence 2, 5, 8, 11, ...

  1. 92
  2. 86
  3. 90
  4. 89 Correct answer.

Explanation

The sequence starts at 2 and increases by 3 each time, so the nth term is 2 + 3(n - 1) = 3n - 1. Substituting n = 30 gives 3(30) - 1 = 89.

Question 20Numbers and Algebraic Expressions

Expand (√5 - 1)².

  1. 4
  2. 6 - 2√5 Correct answer.
  3. 6
  4. 6 + 2√5

Explanation

Use (a - b)² = a² - 2ab + b² with a = √5 and b = 1. This gives 5 - 2√5 + 1 = 6 - 2√5.

Question 21Linear Equations and Functions

Solve the equation 5x + 3 = 2x + 12.

  1. -3
  2. 3 Correct answer.
  3. 9/7
  4. 5

Explanation

Move the x terms to one side and the constants to the other: 5x - 2x = 12 - 3, so 3x = 9. Dividing by 3 gives x = 3.

Question 22Linear Equations and Functions

Solve the simultaneous equations x + y = 10 and x - y = 4. Find the value of x.

  1. 6
  2. 14
  3. 7 Correct answer.
  4. 3

Explanation

Add the two equations to eliminate y: (x + y) + (x - y) = 10 + 4, so 2x = 14 and x = 7. Then y = 10 - 7 = 3, which checks in the second equation: 7 - 3 = 4.

Question 23Linear Equations and Functions

Solve the equation 0.3x - 0.5 = 1.6.

  1. 21
  2. 7 Correct answer.
  3. 2.1
  4. 3.7

Explanation

Add 0.5 to both sides: 0.3x = 2.1. Dividing by 0.3 gives x = 7. Multiplying through by 10 first, 3x - 5 = 16, then 3x = 21, gives the same result.

Question 24Linear Equations and Functions

Solve the inequality -2x + 3 < 9.

  1. x < -3
  2. x > -3 Correct answer.
  3. x > 3
  4. x < 3

Explanation

Subtract 3: -2x < 6. Dividing both sides by the negative number -2 reverses the inequality sign: x > -3.

Question 25Linear Equations and Functions

A shop sells pencils for ¥60 each and pens for ¥80 each. A student buys 18 items in total and pays ¥1,240. How many pens did the student buy?

  1. 12
  2. 7
  3. 10
  4. 8 Correct answer.

Explanation

Let n be the number of pens, so pencils number 18 - n. Then 60(18 - n) + 80n = 1240, which gives 1080 + 20n = 1240, so 20n = 160 and n = 8. Check: 10 pencils cost ¥600 and 8 pens cost ¥640, totaling ¥1,240.

Question 26Linear Equations and Functions

Solve the equation x/3 + 2 = 5.

  1. 9 Correct answer.
  2. 21
  3. 3
  4. 1

Explanation

Subtract 2 from both sides: x/3 = 3. Multiplying both sides by 3 gives x = 9. Check: 9/3 + 2 = 3 + 2 = 5.

Question 27Linear Equations and Functions

Find the larger solution of the quadratic equation x² - 5x - 14 = 0.

  1. -7
  2. 7 Correct answer.
  3. 2
  4. 14

Explanation

Factor the left side: x² - 5x - 14 = (x - 7)(x + 2) = 0, so x = 7 or x = -2. The larger solution is 7.

Question 28Linear Equations and Functions

Solve the quadratic equation x² - 6x + 2 = 0 using the quadratic formula.

  1. 6 ± √7
  2. -3 ± √7
  3. 3 ± 2√7
  4. 3 ± √7 Correct answer.

Explanation

With a = 1, b = -6, c = 2, the formula gives x = (6 ± √(36 - 8))/2 = (6 ± √28)/2. Since √28 = 2√7, this simplifies to (6 ± 2√7)/2 = 3 ± √7.

Question 29Linear Equations and Functions

Five times a certain number minus 8 equals 27. Find the number.

  1. 19/5
  2. 27/5
  3. 7 Correct answer.
  4. 35

Explanation

Let the number be x. Then 5x - 8 = 27, so 5x = 35 and x = 7. Check: 5 × 7 - 8 = 35 - 8 = 27.

Question 30Linear Equations and Functions

A car travels at 60 km/h for 1.5 hours and then at 80 km/h for 0.5 hours. What is its average speed for the whole journey?

  1. 75 km/h
  2. 130 km/h
  3. 70 km/h
  4. 65 km/h Correct answer.

Explanation

Total distance is 60 × 1.5 + 80 × 0.5 = 90 + 40 = 130 km, and total time is 2 hours. Average speed = 130 ÷ 2 = 65 km/h.

Question 31Linear Equations and Functions

How many grams of water must be added to 200 g of an 8% salt solution to dilute it to a 5% salt solution?

  1. 120 g Correct answer.
  2. 320 g
  3. 150 g
  4. 100 g

Explanation

The original solution contains 200 × 0.08 = 16 g of salt, and this does not change. For a 5% solution, 16 g must be 5% of the total, so the total mass is 16 ÷ 0.05 = 320 g. The water added is 320 - 200 = 120 g.

Question 32Linear Equations and Functions

A father is 42 years old and his son is 12 years old. In how many years will the father be exactly twice as old as the son?

  1. 18 Correct answer.
  2. 30
  3. 6
  4. 15

Explanation

Let x be the number of years. Then 42 + x = 2(12 + x), so 42 + x = 24 + 2x, which gives x = 18. Check: in 18 years the father is 60 and the son is 30, and 60 = 2 × 30.

Question 33Linear Equations and Functions

Solve the simultaneous equations 2x + 3y = 12 and x - y = 1. Find the value of x.

  1. 4
  2. 2
  3. 5
  4. 3 Correct answer.

Explanation

From the second equation, x = y + 1. Substituting into the first: 2(y + 1) + 3y = 12, so 5y + 2 = 12, giving y = 2 and x = 3.

Question 34Linear Equations and Functions

How many integers x satisfy both 2x - 1 < 7 and x ≥ 0?

  1. 5
  2. 3
  3. 4 Correct answer.
  4. 7

Explanation

From 2x - 1 < 7, we get 2x < 8, so x < 4. Combined with x ≥ 0, the integers are 0, 1, 2, and 3, which is 4 integers.

Question 35Linear Equations and Functions

Find the smaller solution of the equation (x - 3)(x + 4) = 0.

  1. -4 Correct answer.
  2. -3
  3. 4
  4. 3

Explanation

A product is zero when either factor is zero: x - 3 = 0 gives x = 3, and x + 4 = 0 gives x = -4. The smaller solution is -4.

Question 36Linear Equations and Functions

Solve the equation 7 - 2x = 15.

  1. -4 Correct answer.
  2. -11
  3. 11
  4. 4

Explanation

Subtract 7 from both sides: -2x = 8. Dividing by -2 gives x = -4. Check: 7 - 2(-4) = 7 + 8 = 15.

Question 37Linear Equations and Functions

Pipe A alone can fill a tank in 6 hours, and pipe B alone can fill it in 3 hours. Working together, how long will the two pipes take to fill the tank?

  1. 3 hours
  2. 2 hours Correct answer.
  3. 9 hours
  4. 4.5 hours

Explanation

Pipe A fills 1/6 of the tank per hour and pipe B fills 1/3 per hour. Together they fill 1/6 + 1/3 = 1/6 + 2/6 = 1/2 of the tank per hour, so the tank fills in 2 hours.

Question 38Linear Equations and Functions

The product of two consecutive positive integers is 132. Find the smaller integer.

  1. 12
  2. 66
  3. 10
  4. 11 Correct answer.

Explanation

Let the smaller integer be n. Then n(n + 1) = 132, so n² + n - 132 = 0, which factorizes as (n + 12)(n - 11) = 0. Since n is positive, n = 11. Check: 11 × 12 = 132.

Question 39Linear Equations and Functions

y is directly proportional to x, and y = 12 when x = 4. Find the value of y when x = 7.

  1. 84
  2. 21 Correct answer.
  3. 28
  4. 19

Explanation

Direct proportion means y = ax. From 12 = 4a, the constant is a = 3. When x = 7, y = 3 × 7 = 21.

Question 40Linear Equations and Functions

y is inversely proportional to x, and y = 6 when x = 4. Find the value of y when x = 8.

  1. 12
  2. 2
  3. 3 Correct answer.
  4. 48

Explanation

Inverse proportion means y = a/x, so xy = a. Here a = 4 × 6 = 24. When x = 8, y = 24 ÷ 8 = 3.

Question 41Linear Equations and Functions

Find the slope of the line passing through the points (1, 2) and (3, 8).

  1. 3 Correct answer.
  2. 6
  3. 1/3
  4. 2

Explanation

Slope = (change in y) ÷ (change in x) = (8 - 2) ÷ (3 - 1) = 6 ÷ 2 = 3.

Question 42Linear Equations and Functions

Find the equation of the line with slope 2 that passes through the point (1, 5).

  1. y = 5x + 2
  2. y = 2x - 3
  3. y = 2x + 5
  4. y = 2x + 3 Correct answer.

Explanation

Write y = 2x + b and substitute the point: 5 = 2(1) + b, so b = 3. The equation is y = 2x + 3.

Question 43Quadratic Functions and Equations

For the function y = ax², when x = 2 the value is y = 12. Find the value of a.

  1. 3 Correct answer.
  2. 4
  3. 24
  4. 6

Explanation

Substitute the values: 12 = a × 2² = 4a. Solving gives a = 3.

Question 44Quadratic Functions and Equations

For the function y = 3x², find the rate of change (average rate of change) as x increases from 2 to 4.

  1. 6
  2. 9
  3. 18 Correct answer.
  4. 36

Explanation

When x = 2, y = 12; when x = 4, y = 48. The rate of change is (48 - 12) ÷ (4 - 2) = 36 ÷ 2 = 18. Equivalently, for y = ax² the rate from p to q is a(p + q) = 3(2 + 4) = 18.

Question 45Linear Equations and Functions

Which of the following points lies on the graph of y = -x + 4?

  1. (3, -1)
  2. (3, 1) Correct answer.
  3. (2, 6)
  4. (1, 5)

Explanation

Substitute x = 3 into the equation: y = -3 + 4 = 1. So the point (3, 1) lies on the graph.

Question 46Linear Equations and Functions

Find the coordinates of the intersection point of the lines y = x + 1 and y = -2x + 7.

  1. (2, 5)
  2. (-2, 3)
  3. (3, 2)
  4. (2, 3) Correct answer.

Explanation

Set the expressions equal: x + 1 = -2x + 7, so 3x = 6 and x = 2. Substituting back gives y = 2 + 1 = 3, so the intersection is (2, 3).

Question 47Linear Equations and Functions

For the function y = 3x with domain -1 ≤ x ≤ 2, find the range of y.

  1. -1 ≤ y ≤ 6
  2. -3 ≤ y ≤ 6 Correct answer.
  3. 3 ≤ y ≤ 6
  4. -3 ≤ y ≤ 2

Explanation

Since the slope is positive, y is smallest at x = -1 and largest at x = 2. At x = -1, y = -3; at x = 2, y = 6. The range is -3 ≤ y ≤ 6.

Question 48Quadratic Functions and Equations

For the function y = ax², the rate of change as x increases from 1 to 4 is 10. Find the value of a.

  1. 2 Correct answer.
  2. 5
  3. 5/2
  4. 10/3

Explanation

The rate of change of y = ax² from x = p to x = q is a(p + q). Here a(1 + 4) = 10, so 5a = 10 and a = 2. Check: y goes from 2 to 32, and (32 - 2) ÷ (4 - 1) = 30 ÷ 3 = 10.

Question 49Linear Equations and Functions

A tank holds 50 L of water and drains at a steady rate of 2 L per minute. After how many minutes will 20 L remain in the tank?

  1. 10 minutes
  2. 25 minutes
  3. 35 minutes
  4. 15 minutes Correct answer.

Explanation

The amount remaining after x minutes is y = 50 - 2x. Setting 50 - 2x = 20 gives 2x = 30, so x = 15 minutes.

Question 50Linear Equations and Functions

Find the equation of the line that passes through the point (2, 3) and is parallel to the line y = 4x - 1.

  1. y = 2x - 1
  2. y = 4x + 3
  3. y = 4x - 5 Correct answer.
  4. y = 4x - 1

Explanation

Parallel lines have the same slope, so the line is y = 4x + b. Substituting (2, 3): 3 = 4(2) + b, so b = 3 - 8 = -5. The equation is y = 4x - 5.