Question
The length, pcm, of a car is 440cm, correct to the nearest 10cm.
Complete the statement about p.
▶️Answer/Explanation
The value of p lies in the range 435 ≤ p < 445.The value 435 represents the lower bound of the range, indicating that the car’s length can be equal to or greater than 435 centimeters. The value 445 represents the upper bound of the range, indicating that the car’s length must be strictly less than 445 centimeters.Therefore, any value of p within the range 435 ≤ p < 445, such as 436, 439.5, or 444, would be considered valid or correct based on the given statement.
Question
Which two of these have the same value?
\(5^{-2}\) \(\frac{2}{5}\) \((\frac{1}{2})^2\) \((\frac{2}{5})^2\) \(0.2^2\)
(b) Simplify.
(i) \(a^6 \times a^3\)
(ii) \(24b^{16} \div 6b^4\)
▶️Answer/Explanation
(a)We can simplify each expression to compare them,
\(5^{-2}=\frac{1}{5^{2}}=\frac{1}{25}\)
\(\frac{2}{5}=0.4\)
\(\left ( \frac{1}{2} \right )^{2}=\frac{1^{2}}{2^{2}}=\frac{1}{4}\)
\(\left ( \frac{2}{5} \right )^{2}=\frac{2^{2}}{5^{2}}=\frac{4}{25}\)
\(0.2^{2}=0.04\)
From the above calculations, we can see that \(\left ( \frac{2}{5} \right )^{2}\) is equal to \(5^{-2} \)since they both equal \(\frac{4}{25}\).Therefore, \(\left ( \frac{2}{5} \right ) ^{2}\) and \(5^{-2}\) have the same value.
(b) (i) When multiplying two terms with the same base, we can add their exponents.So,
\(a^{6}\times a^{3}=a^{6+3}=a^{9}\)
Therefore,\(a^{6}\times a^{3} \)is equal to\( a^{9}.\)
(ii) To simplify the expression \(24b^{16}\div 6b^{4}\), we can simplify the numerator and denominator separately and then divide:
\(24b^{16}\div 6b^{4}=\frac{24}{6}\times \frac{b^{16}}{b^{4}}=4b^{16-4}=4b^{12}.\)
Therefore,\(24b^{16}\div 6b^{4} \)simplifies to \(4b^{12}.\)
Question
x and y are integers.
(a) Find the value of x when –7 < x < –5.
(b) Find the value of y when \(\frac{3}{4} < \frac{y}{16} < \frac{7}{8}\).
▶️Answer/Explanation
(a) The value of x must be an integer that satisfies the inequality –7 < x < –5.This means that x must be an integer between -7 and -5 ,greater than -7 and less than -5 which is -6.
(b) To solve the inequality \(\frac{3}{4}< \frac{y}{16}< \frac{7}{8}\), we can multiply all sides of the inequality by 16. Since we know that 16 is a positive number and therefore multiplying by 16 will not change the direction of the inequality.
\(\therefore \frac{3}{4}\times 16< y< \frac{7}{8}\times 16\)
Simplifying each side of the inequality gives 12< y< 14
Therefore, the possible values of y that satisfy the inequality are integer between 12 and 14 that is 13.
So,the value of y when \(\frac{3}{4} < \frac{y}{16} < \frac{7}{8}\) is 13
Question
Sebastian ran a race in 11.4 seconds, correct to 1 decimal place.
Complete the statement about the time, t seconds, that Sebastian took to run the race.
▶️Answer/Explanation
Since Sebastian’s race time was given as 11.4 seconds, correct to 1 decimal place, we know that the actual time t must be between 11.35 seconds and 11.45 seconds.This is because when we round 11.4 to 1 decimal place, we round to the nearest tenth, which means the last digit (in this case, 4) could have been rounded up from a value between 0 and 4, or rounded down from a value between 5 and 9.
Therefore,11.35< t< 11.45
So the time, t seconds, that Sebastian took to run the race is between 11.35 seconds and 11.45 seconds.
Question
Write the following in order of size, smallest first.
π 3.14 \(\frac{22}{7}\) 3.142 3
▶️Answer/Explanation
π (pi) is a mathematical constant approximately equal to 3.14159. In the given list, it is represented as “π”. It is commonly approximated as 3.14.
3.14 is a decimal approximation of pi (π). It is less precise than the actual value of pi and therefore larger than pi itself.
\frac{22}{7} is a common fractional approximation of pi (π). While it is not equal to the exact value of pi, it provides a reasonably accurate approximation.
3.142 is another decimal approximation of pi (π), which is slightly more precise than 3.14 but still smaller.
3 is an integer, representing a whole number without any decimal part. It is the largest value in the given list.
Therefore, the values in order of size, from smallest to largest, would be \(3< 3.14< \frac{22}{7}< 3.142< \pi\)