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Write each number in prime factors: 12 = 2² × 3 and 18 = 2 × 3². The LCM takes the highest power of every prime, so 2² × 3² = 36. The LCM must be a multiple of both numbers and at least as large as the larger one, which rules out 6.
Rewrite both fractions with the common denominator 12: 2/3 = 8/12 and 1/4 = 3/12. Adding the numerators gives 8 + 3 = 11, so the sum is 11/12. The denominators are not added.
The decimal 0.25 means 25 hundredths, that is, 25/100. Dividing numerator and denominator by 25 gives 1/4. Since 0.25 is less than 1, the fraction must be less than 1.
15% means 15 per 100, so 15% of 200 = (15/100) × 200. This equals 15 × 2 = 30.
All three terms contain the same variable x, so they are like terms. Add and subtract the coefficients: 3 + 5 − 2 = 6. The result is 6x.
Subtract 7 from both sides to isolate x: x = 12 − 7. Therefore x = 5. Check: 5 + 7 = 12.
The area of a rectangle is length × width. Here 8 × 5 = 40 square units.
The mean is the sum of the values divided by how many there are. The sum is 4 + 6 + 8 + 10 = 28, and 28 ÷ 4 = 7.
The GCD is the product of the common prime factors with the lowest powers. 24 = 2³ × 3 and 36 = 2² × 3², so the common part is 2² × 3 = 12.
The ratio 3 : 5 has 3 + 5 = 8 equal parts. One part is 40 ÷ 8 = 5 students. Girls are 5 parts, so 5 × 5 = 25.
When multiplying powers with the same base, add the exponents: 2³ × 2⁴ = 2^(3+4) = 2⁷. Since 2⁷ = 128, the answer is 128.
Add 5 to both sides: 3x = 21. Divide both sides by 3: x = 7. Check: 3(7) − 5 = 16.
√169 = 13 because 13 × 13 = 169. √49 = 7 because 7 × 7 = 49. Therefore 13 − 7 = 6.
The expression is a difference of two squares: a² − b² = (a − b)(a + b). Here a = x and b = 3, so x² − 9 = (x − 3)(x + 3).
There are 1000 metres in a kilometre. Multiply by 1000: 2.5 × 1000 = 2500 metres. Moving from a larger unit to a smaller unit makes the number larger.
A fair die has six equally likely outcomes: 1, 2, 3, 4, 5, 6. The even outcomes are 2, 4 and 6, so there are 3 favourable outcomes. Probability = 3/6 = 1/2.
Arrange the values in order: 3, 4, 5, 7, 9. With five values, the median is the third one. Therefore the median is 5.
The angles of a triangle add up to 180°. The two given angles sum to 50° + 60° = 110°. Therefore the third angle is 180° − 110° = 70°.
Complementary angles add up to 90°. The complement of 35° is 90° − 35° = 55°.
Divide both sides by 2: x + 3 = 7. Then subtract 3: x = 4. Check: 2(4 + 3) = 14.
Adding the two equations eliminates y: (x + y) + (x − y) = 10 + 4, so 2x = 14 and x = 7. Substituting into x + y = 10 gives y = 3. Then xy = 7 × 3 = 21.
To divide by a fraction, multiply by its reciprocal: 3/4 ÷ 9/8 = 3/4 × 8/9. Multiply numerators and denominators: (3 × 8)/(4 × 9) = 24/36. Simplify by dividing by 12 to get 2/3.
The area of a circle is A = πr². With r = 7 and π ≈ 22/7, A = (22/7) × 49 = 22 × 7 = 154 cm².
Divide both sides by −2, and reverse the inequality sign because dividing by a negative number reverses it. This gives x > −3. Checking x = 0 gives 0 < 6, which is true.
After a 20% increase the amount is 1.20 times the original. A 20% decrease then multiplies by 0.80, giving 1.20 × 0.80 = 0.96, which is 96% of the original. So the net result is a 4% decrease.
Like terms in m can be combined by adding their coefficients: 7 − 3 + 2 = 6. Therefore the simplified expression is 6m.
The area of a triangle is one half of the base multiplied by the perpendicular height. Thus area = 1/2 × 12 × 7 = 42 cm².
0.75 means 75 hundredths, so 0.75 = 75/100. Dividing numerator and denominator by 25 gives 3/4, which is simplest form.
To round to the nearest hundred, look at the tens digit. The tens digit is 3, which is less than 5, so keep the hundreds digit and change the last two digits to zero. The result is 4,700.
Write 12 = 2² × 3 and 18 = 2 × 3². The LCM uses the highest power of each prime factor: 2² × 3² = 4 × 9 = 36.
A percentage of an amount is found by multiplying by the percentage divided by 100. Thus 15% of 240 = 0.15 × 240 = 36.
Move the decimal point 4 places to the right to get 4.2, so multiply by 10⁻⁴. Standard scientific notation uses a coefficient between 1 and 10, giving 4.2 × 10⁻⁴.
Multiply the coefficients: 2 × 3 = 6. When multiplying powers with the same base, add the exponents: a² × a³ = a⁵, so the result is 6a⁵.
Angles on a straight line add up to 180°. The other angle is 180° − 118° = 62°.
There are 3 red balls out of a total of 3 + 5 = 8 balls. Probability = favourable outcomes / total outcomes = 3/8.
Use Pythagoras’ theorem: hypotenuse² = 6² + 8² = 36 + 64 = 100. Therefore the hypotenuse is √100 = 10 cm.
The midpoint coordinates are the averages of the x-coordinates and y-coordinates. x = (2+8)/2 = 5 and y = (5+1)/2 = 3, so the midpoint is (5, 3).
We need two numbers whose product is 6 and whose sum is 5. These numbers are 2 and 3, so x² + 5x + 6 = (x + 2)(x + 3).
The exterior angles of any polygon add up to 360°. For a regular polygon, the number of sides is 360° ÷ exterior angle = 360° ÷ 24° = 15.
Simple interest = principal × rate × time = 5000 × 0.08 × 3 = 1200. The total amount is principal plus interest: 5000 + 1200 = 6200.
The subject of the sentence is 'each', a singular indefinite pronoun, so the verb must take the singular form 'has'. The phrase 'of the students' is only a prepositional modifier and does not change the number of the subject. Plural and continuous forms do not agree with a singular indefinite pronoun, and the sentence describes a simple present fact rather than an action in progress.
A singular countable noun introduced for the first time normally takes an indefinite article. The choice between 'a' and 'an' depends on the sound that follows: the 'h' in 'honest' is silent, so the word starts with a vowel sound and requires 'an'. Because 'man' is countable and singular, the article cannot be left out.
'Child' is one of the irregular nouns that form the plural by changing the vowel and adding '-en' instead of '-s'. The correct plural is therefore 'children'. Adding a regular plural ending to an irregular form, or adding a second plural ending, is never correct in standard English.
'Since' is used with a point in time to show when an action that continues began, and it pairs naturally with the present perfect continuous. 'For' introduces a length of time such as 'for nine years', and 'during' is followed by a period or event. 'From' normally needs a second point, as in 'from 2015 to 2020', to express a range.
A possessive pronoun stands alone and replaces a possessive adjective plus its noun, so 'mine' means 'my pencil'. 'My' is a possessive adjective and must be followed by a noun. 'Me' is an object pronoun and 'myself' is reflexive, so neither of them can express ownership at the end of the sentence.
'Ancient' describes something very old, so its opposite must refer to the present or to recent times. 'Modern' is the direct antonym and is the natural pairing in expressions such as 'ancient and modern buildings'. The other options relate to historical importance, size or fame, but none of them expresses age.
The time expression 'every day' signals a habitual action, which requires the simple present tense. With a third-person singular subject such as 'she', the verb takes the '-s' ending, giving 'goes'. The remaining forms are a base form, a present participle and a past participle, none of which can act as a finite present-tense verb here.
When a subordinate clause beginning with a conjunction such as 'although' comes before the main clause, a comma separates the two clauses. Here the comma must fall after 'raining', where the dependent clause ends. Placing a comma straight after the conjunction, or in the middle of the phrasal verb 'went out', breaks the sentence into meaningless pieces.
The phrase 'for ten years' describes a period that began in the past and continues up to the present, which requires the present perfect tense. The past participle 'known' must therefore be preceded by 'have'. The simple past would need a finished time expression, and 'know' is a stative verb that is not normally used in the continuous form.
In a passive sentence the receiver of the action becomes the subject, so the verb must take the pattern 'be + past participle'. The time word 'yesterday' places the action in the past, which makes 'was written' the correct past simple passive. The active forms would require the secretary to be the subject of the sentence.