Luyện Toán GCSE

Use a common denominator of 20: 3/5 = 12/20 and 1/4 = 5/20, so 12/20 + 5/20 = 17/20.

20% = 0.2, so 0.2 x 45 = 9. To find a percentage, convert it to a decimal and multiply by the original amount. Checking 10% of 45 = 4.5 also confirms that 20% is double 4.5, which is 9.

25% of 80 is 20, so 80 + 20 = 100. An increase means adding the percentage amount to the original value. Since one quarter of 80 is 20, the increased value is 80 + 20 = 100.

Multiply both parts by 4: 3:5 = 12:20. Equivalent ratios are formed by multiplying or dividing both parts by the same number. Because 3 x 4 = 12 and 5 x 4 = 20, 12:20 keeps the same relationship as 3:5.

Multiples of 6 include 6, 12, 18, 24; multiples of 8 include 8, 16, 24, so the LCM is 24.

The common factors of 12 and 18 are 1, 2, 3 and 6, so the highest is 6.

Move the decimal point 4 places to the right to get 7.2, so the power is -4.

12 x 12 = 144, so sqrt(144) = 12. The positive square root is the nonnegative number that squares to give the original value. Squaring 12 confirms 144, while 14 x 14 = 196 and 11 x 11 = 121.

2^5 = 2 x 2 x 2 x 2 x 2 = 32.

Add the powers: 3^2 x 3^3 = 3^5 = 243.

Any non-zero number raised to the power 0 is 1.

Collect like terms: 6x - 2x = 4x and 3 + 7 = 10, so the result is 4x + 10.

Subtract 7 from both sides: x = 15 - 7 = 8. Solving x + 7 = 15 requires isolating x by doing the inverse operation on both sides. Substituting x = 8 back gives 8 + 7 = 15, which checks.

Divide both sides by 3: x = 21 / 3 = 7. Because x is multiplied by 3, the inverse operation is division. Checking 3 x 7 = 21 confirms that 7 is the correct solution.

Subtract 5 then divide by 2: 2x = 8, so x = 4.

Multiply both sides by 4: x = 24. Because x is divided by 4, the inverse operation is multiplication. Substituting 24 back gives 24 / 4 = 6, so the solution checks.

Subtract 2x from both sides to get 3x - 3 = 9, then add 3: 3x = 12, so x = 4.

Multiply each term inside the bracket by 3: 3x + 12.

Using FOIL: x^2 + 5x + 2x + 10 = x^2 + 7x + 10.

The common factor is x, so x^2 + 5x = x(x + 5).

Divide coefficients and subtract powers: 10/2 = 5 and 5 - 2 = 3, so 5x^3.

The sequence increases by 4 each time, so the next term is 15 + 4 = 19.

The difference is 3, so the nth term has 3n; at n = 1, 3n = 3 and 3 + 2 = 5, so it is 3n + 2.

Add 4 to both sides: 2x > 10, then divide by 2: x > 5.

In y = mx + c, m is the gradient, so the gradient is 3.

In y = mx + c, c is the y-intercept, so the y-intercept is 7.

Substitute x = 4: y = 2(4) + 1 = 9. Replace x with 4, multiply by 2 first, then add 1. The result is 8 + 1 = 9, so the point (4, 9) lies on the line.

Add the equations: 2x = 14, so x = 7. Then y = 3.

Add the equations: 3x = 15, so x = 5. Substitute into x - y = 3 to get y = 2.

x^2 = 9, so x = 3 or x = -3. The positive solution is 3.

The interior angles of a triangle always add up to 180 degrees.

A quadrilateral can be split into two triangles, so 2 x 180 = 360 degrees.

Area = length x width = 6 x 4 = 24 cm^2.

Area = 1/2 x base x height = 1/2 x 10 x 6 = 30 cm^2.

Area = pi r^2 = pi x 3^2 = 9 pi cm^2.

Circumference = 2 pi r = 2 x pi x 5 = 10 pi cm.

Using Pythagoras: c^2 = 3^2 + 4^2 = 9 + 16 = 25, so c = 5 cm.

Using Pythagoras: 13^2 - 5^2 = 169 - 25 = 144, so the missing side is sqrt(144) = 12 cm.

Volume = length x width x height = 2 x 3 x 4 = 24 cm^3.

Volume = pi r^2 h = pi x 2^2 x 5 = 20 pi cm^3.

Perimeter = 4 x side, so side = 28 / 4 = 7 cm.

Area = base x height = 8 x 5 = 40 cm^2.

Mean = (4 + 6 + 8 + 10 + 12) / 5 = 40 / 5 = 8.

The middle value of the ordered list is 7. With five numbers, the median is the third value once they are arranged from smallest to largest. The list is already ordered, so the middle number is 7; 5 is the first value and 9 is the fourth.

The mode is the value that appears most often, which is 3.

Range = largest - smallest = 12 - 2 = 10. The range measures how spread out a data set is by subtracting the smallest value from the largest. For 2, 5, 9 and 12, the spread is 12 - 2 = 10.

There are 3 even numbers out of 6, so P(even) = 3/6 = 1/2.

One outcome is a 6 out of six equally likely outcomes, so P(6) = 1/6.

The outcomes are HH, HT, TH and TT, so P(HH) = 1/4.

P(blue) = 12/30 = 2/5. Probability is the number of favorable outcomes divided by the total number of equally likely outcomes. Simplify 12/30 by dividing both numbers by 6 to get 2/5.

There are 10 counters in total and 3 are red, so P(red) = 3/10.

The two probabilities add to 1, so P(no rain) = 1 - 0.3 = 0.7.

5% of 200 is 10, so the total after one year is 200 + 10 = 210 pounds.

Interest = principal x rate x time = 500 x 0.04 x 2 = 40 pounds.

The decrease is 20, so 20/80 = 0.25 = 25%.

A negative power gives a reciprocal: 4^-1 = 1/4.

0.3 x 0.3 = 0.09, so sqrt(0.09) = 0.3.

This is the difference of two squares: x^2 - 25 = (x - 5)(x + 5).

Factorise to (x - 2)(x - 5) = 0, so x = 2 or x = 5. The smaller solution is 2.

Gradient = (7 - 1) / (2 - 0) = 6 / 2 = 3. The gradient measures the change in y divided by the change in x between two points. Moving from (0, 1) to (2, 7) rises 6 units over a run of 2, so the gradient is 3.

A prime number has exactly two distinct factors: 1 and itself.

3^2 = 3 x 3 = 9, not 6. The exponent 2 means multiply the base by itself once, so 3 squared is 9. Multiplying 3 by 2 gives 6, which confuses powers with multiplication.

The interior angles of any triangle sum to 180 degrees.

The middle value is the median; the mean is the sum divided by the number of values.

Probabilities are always between 0 and 1 inclusive. An event that cannot happen has probability 0, and a certain event has probability 1. A value greater than 1 would mean more favorable outcomes than total outcomes, which is impossible.

Multiply both terms by 2: 2x + 6. Expanding a bracket means applying the outside factor to every term inside. Since 2(x + 3) = 2x + 6, the statement is true.

A square has four right angles, so it satisfies the definition of a rectangle.

1 divided by 2 is 0.5, so the values are equal.

2 and 6 divide 12 exactly; 5 does not and 24 is a multiple, not a factor.

2 + 8 = 10 and 20 / 2 = 10; the other two equal 9.

7 and 11 have exactly two factors; 9 = 3 x 3 and 15 = 3 x 5 are not prime.

A square and a rhombus have four equal sides; a rectangle does not and a kite has two pairs of adjacent equal sides.

4 x 4 = 16 and (-4) x (-4) = 16; 8^2 and (-8)^2 are 64.

12, 18 and 30 are in the 6 times table; 22 is not divisible by 6.

A prime has exactly two factors, a factor divides exactly, a multiple is in a times table, and a square number is an integer times itself.

A coefficient multiplies a variable, a variable is an unknown, an expression has no equals sign, and an equation states equality.

The radius reaches the circumference, the diameter crosses the centre, a chord joins two points, and a tangent touches at one point.

Mean is the average, median is the middle, mode is most frequent, and range measures spread.

Impossible events have probability 0, certain events have 1, an even chance is 1/2, and unlikely events fall below 1/2.

A reflection mirrors, a rotation turns, a translation slides, and an enlargement changes size.