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In the number 3,742, the digit 7 is in the hundreds place, so its value is 7 × 100 = 700. The place values from right to left are ones, tens, hundreds, thousands.
A prime number has exactly two distinct factors: 1 and itself. 29 is not divisible by 2, 3, or 5, and its only factors are 1 and 29. The other numbers are all divisible by 3.
Multiply the digits: 6 × 4 = 24. There is one decimal place in 0.6 and one in 0.4, giving two decimal places in the product, so the answer is 0.24.
To convert 3/5 to a decimal, divide 3 by 5. Since 3 ÷ 5 = 0.6, the decimal form is 0.6. Alternatively, multiply numerator and denominator by 2 to get 6/10 = 0.6.
The factors of 18 are 1, 2, 3, 6, 9, 18; the factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The largest factor common to both lists is 6.
Using the order of operations, evaluate the power first: 2³ = 8. Then multiply: 4 × 2 = 8. Finally add: 8 + 8 = 16.
Each term is obtained by multiplying the previous term by 3: 2 × 3 = 6, 6 × 3 = 18, 18 × 3 = 54. Therefore the next term is 54 × 3 = 162.
A number is divisible by 9 if the sum of its digits is divisible by 9. For 234, 2 + 3 + 4 = 9, which is divisible by 9, so 234 is divisible by 9.
The amount of flour per cake is 250 ÷ 10 = 25 g. For 25 cakes, the flour needed is 25 × 25 = 625 g. Alternatively, 25 cakes is 2.5 times 10 cakes, so 2.5 × 250 = 625 g.
Standard form is a × 10ⁿ where 1 ≤ a < 10. Moving the decimal point in 0.00045 four places to the right gives 4.5, so the exponent is −4. Thus 0.00045 = 4.5 × 10⁻⁴.
Multiply the coefficients: 2 × 3 = 6. Add the indices: 10⁵ × 10⁻² = 10³. The result is 6 × 10³, which is already in standard form.
The reduction is 15% of $40 = 0.15 × 40 = $6. Subtract this from the original price: $40 − $6 = $34. Alternatively, the sale price is 85% of $40 = 0.85 × 40 = $34.
Expand each bracket: 3(2x + 4) = 6x + 12 and −2(x − 5) = −2x + 10. Then combine like terms: 6x − 2x = 4x and 12 + 10 = 22, giving 4x + 22.
Subtract 2x from both sides: 3x − 3 = 12. Add 3 to both sides: 3x = 15. Divide by 3: x = 5.
The total number of parts is 3 + 5 = 8. Each part represents 40 ÷ 8 = 5 students. The number of girls is 5 parts × 5 = 25.
Average speed is distance divided by time: 150 km ÷ 2 h = 75 km/h. This means the car covers 75 km each hour on average.
To round to 2 decimal places, look at the third decimal digit, which is 6. Since 6 is 5 or more, increase the second decimal digit (4) by 1, giving 7.85.
If the original number is x, then 1.2x = 72. Dividing both sides by 1.2 gives x = 60. Check: 20% of 60 is 12, and 60 + 12 = 72.
First apply the power of a power rule: (2³)² = 2^(3×2) = 2⁶. Then multiply by 2⁻¹ by adding indices: 2⁶ × 2⁻¹ = 2^(6−1) = 2⁵.
Divide 360 repeatedly by prime numbers: 360 = 2 × 180 = 2² × 90 = 2³ × 45. Then 45 = 3 × 15 = 3² × 5. So 360 = 2³ × 3² × 5.
Simple interest is I = PRT/100 = (2000 × 5 × 3)/100 = $300. The total amount is the principal plus interest: $2000 + $300 = $2300.
The increase is 50 − 35 = 15°C. The percentage increase is (increase ÷ original) × 100 = (15 ÷ 35) × 100 ≈ 42.9%.
Divide the coefficients: 2 ÷ 4 = 0.5. Subtract the indices: 10³ ÷ 10⁻² = 10^(3−(−2)) = 10⁵. So x ÷ y = 0.5 × 10⁵ = 5 × 10⁴ in standard form.
Let the principal be P. After 2 years, P × (1.04)² = 2704. Since 1.04² = 1.0816, P = 2704 ÷ 1.0816 = 2500. Check: 2500 × 1.04² = 2704.
Move the decimal point four places to the right to get 3.06, so the exponent is −4. The trailing zero after 6 is not significant in this context, so the standard form is 3.06 × 10⁻⁴. Option 1 is numerically the same but includes a non-significant trailing zero, which is not usual in standard form.
An irrational number cannot be expressed as a fraction of two integers and its decimal expansion is non-terminating and non-recurring. √2 is the classic example, while √9 = 3, 0.333... = 1/3, and 22/7 is a rational approximation of π, hence all are rational.
In the decimal 3.472, the digits after the decimal point represent tenths, hundredths, and thousandths in order. The 4 is in the tenths place, the 7 is in the hundredths place, and the 2 is in the thousandths place, so the 7 represents 7 hundredths.
The order of operations requires evaluating indices (powers) before multiplication and division, and those before addition and subtraction. So first square 4 to get 16, then multiply by 3 to get 48, and finally add 2 to obtain 50.
The highest common factor is the largest number that divides exactly into both 24 and 36. Listing factors: 24 has factors 1,2,3,4,6,8,12,24; 36 has 1,2,3,4,6,9,12,18,36. The largest common factor is 12.
A prime factorisation expresses the number as a product of prime numbers only. Starting with 84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7, we get 2² × 3 × 7, which are all primes.
0.75 means 75 hundredths, so it can be written as 75/100. Dividing numerator and denominator by 25 gives 3/4, which is the simplest form.
The exponent −3 means the decimal point in 3.6 is moved three places to the left. Starting with 3.6, moving one place gives 0.36, two places gives 0.036, and three places gives 0.0036.
Dividing by a fraction is the same as multiplying by its reciprocal. So (2/3) ÷ (4/9) = (2/3) × (9/4) = 18/12 = 3/2. The answer is already in simplest form.
Let x = 0.333..., then 10x = 3.333... Subtracting gives 9x = 3, so x = 3/9 = 1/3. This is the exact fraction for the recurring decimal.
A prime number has exactly two distinct factors: 1 and itself. Between 20 and 30, the numbers 23 and 29 are prime; 21, 25 and 27 are composite, and 22, 24, 26, 28 are even.
A negative index means the reciprocal of the positive power: 2⁻³ = 1/(2³) = 1/8. The result is positive because the base 2 is positive.
The ratio 5:2 means for every 5 parts flour there are 2 parts sugar. One part is 350 ÷ 5 = 70 g, so sugar is 2 × 70 = 140 g.
A 15% reduction means the customer pays 85% of the original price. 85% of $80 is 0.85 × 80 = $68.
Multiply numerators and denominators: (3×8)/(4×9) = 24/36. Simplifying by dividing both by 12 gives 2/3.
Standard form is a × 10ⁿ where 1 ≤ a < 10. Moving the decimal point 4 places to the right gives 4.5, so the index is −4: 4.5 × 10⁻⁴.
The index −2 means the digits 5.04 must be divided by 10 twice, so the decimal point moves two places to the left from 5.04. Moving two places gives 0.0504. This is a small number, which is consistent with a negative index.
Fuel consumption in km per litre is found by dividing distance by litres: 240 ÷ 18 = 13.333… Rounding to 3 significant figures gives 13.3, since the fourth digit is 3 and so the third digit stays the same.
The difference between Chen's share and Amara's share is 7 − 3 = 4 parts, and this equals $96, so one part is $24. The total is 3 + 5 + 7 = 15 parts, giving 15 × $24 = $360. Checking: Amara gets $72, Ben $120 and Chen $168, and indeed 168 − 72 = 96.
N = 16 × 9 × 5 = 720. Since 45 = 9 × 5 and 16 both divide 720, N is a multiple of each of them. The other statements fail: the index of 5 is 1 (odd), so N is not a perfect square, and 720 is divisible by 9.
Collect like terms separately: 5a - 2a = 3a and 3b + 7b = 10b. Unlike terms cannot be combined further, so the simplified expression is 3a + 10b.
Multiply each term inside the bracket by 3: 3 × 2x = 6x and 3 × (-5) = -15. The expansion is therefore 6x - 15.
Add 7 to both sides to get 4x = 28, then divide both sides by 4 to obtain x = 7. Substituting back gives 28 - 7 = 21, which checks.
The highest common factor of 6x and 9 is 3. Dividing each term by 3 gives 2x + 3, so the factorised form is 3(2x + 3).
In the form y = mx + c, the coefficient m is the gradient and c is the y-intercept. Here m = 3, so the gradient is 3, while -4 is the intercept.
When multiplying powers with the same base, add the indices: 3 + 4 = 7, giving 2^7. The base stays the same and the indices are added, not multiplied.