1) Simplify: [tex]$0.0216 \times 10^{-3} \times 0.024 \times 10^{-4} \times 0.018 \times 10^{-5}$[/tex] / [tex]$3600 \times 10^{-6} \times 0.96 \times 10^{-3}$[/tex]
2) The first, second, and third terms of an arithmetic progression are [tex]$a, b, \frac{3}{4}$[/tex]. Find the values of [tex]$a$[/tex] and [tex]$b$[/tex].
3) A trader buys 200 oranges for $1420. She later realizes that 10% of the oranges are rotten and cannot be sold. She sells the remaining oranges at a rate of 3 for $f. Calculate, correct to US$, the true percentage profit or loss.
4) Solve the inequality: [tex]$\frac{x}{3} \geqslant 5-4\left(x-\frac{1}{6}\right)$[/tex]
Answer (2)
Problem 1 simplifies a complex arithmetic expression with scientific notation.
Problem 2 identifies an arithmetic progression but lacks sufficient information for a unique solution.
Problem 3 calculates the percentage profit or loss for a trader selling oranges.
Problem 4 solves a linear inequality for x, resulting in x ⩾ 13 21 .
Explanation
Solving the Problems We will solve the four problems separately.
Problem 1: Evaluate the expression 3600 × 1 0 − 6 × 0.96 × 1 0 − 3 0.0216 × 1 0 − 3 × 0.024 × 1 0 − 4 × 0.018 × 1 0 − 5 First, we calculate the product of the coefficients in the numerator: 0.0216 × 0.024 × 0.018 = 0.0000093312 = 9.3312 × 1 0 − 6 Next, we calculate the product of the coefficients in the denominator: 3600 × 0.96 = 3456 Now, we calculate the product of the powers of 10 in the numerator: 1 0 − 3 × 1 0 − 4 × 1 0 − 5 = 1 0 − 12 Next, we calculate the product of the powers of 10 in the denominator: 1 0 − 6 × 1 0 − 3 = 1 0 − 9 So the expression becomes: 3456 × 1 0 − 9 9.3312 × 1 0 − 6 × 1 0 − 12 = 3456 × 1 0 − 9 9.3312 × 1 0 − 18 = 3456 9.3312 × 1 0 − 9 3456 9.3312 = 0.0027 So the final answer is: 0.0027 × 1 0 − 9 = 2.7 × 1 0 − 12
Problem 2: Given an arithmetic progression (AP) with the first three terms as a, b, c with c = 3/4, find the values of a and b. In an AP, 2b = a + c. We need more information to solve this problem. The problem states 'b, b, b, 3/4' which is incorrect. Assuming the AP is a, b, 3/4, we have 2b = a + 3/4. We cannot uniquely determine a and b with only this information. We will assume that the first term is 'a' and the common difference is 'd'. Then a = a, b = a + d, c = a + 2d = 3/4. So a + 2d = 3/4. We still need more information to solve for a and d. Let's assume a = 0. Then 2d = 3/4, so d = 3/8. Then a = 0, b = 3/8, c = 3/4. However, without more information, we cannot determine unique values for a and b.
Problem 3: A trader buys 200 oranges for 1420.10% o f t h eor an g es a rero tt e nan d c ann o t b eso l d . T h ere mainin g or an g es a reso l d a t a r a t eo f 3 f or f$. Calculate the percentage profit or loss. Number of rotten oranges = 10% of 200 = 20 Number of oranges that can be sold = 200 - 20 = 180 Number of sets of 3 oranges = 180 / 3 = 60 Revenue from selling the oranges = 60f Cost price = 1420 P ro f i t / L oss = 60 f − 1420 P erce n t a g e p ro f i t / l oss = \frac{60f - 1420}{1420} \times 100$
If we assume f = $14.2, then the revenue is 60 * 14.2 = $852. The loss is 1420 - 852 = $568. The percentage loss is (568/1420) * 100 = 40%.
Problem 4: Solve the inequality 3 x ⩾ 5 − 4 ( x − 2 1 ) .
3 x ⩾ 5 − 4 x + 2 3 x ⩾ 7 − 4 x x ⩾ 21 − 12 x 13 x ⩾ 21 x ⩾ 13 21
Examples
These problems cover basic arithmetic, algebra, and financial calculations. For example, calculating profit/loss percentages is crucial in business to assess financial performance. Solving inequalities helps determine the range of possible solutions in various scenarios, such as resource allocation or setting constraints in optimization problems. Understanding arithmetic progressions is useful in predicting patterns and sequences, such as in finance or physics.
The problems include simplifying an arithmetic expression with scientific notation, solving an arithmetic progression, calculating profit/loss from selling oranges, and solving an inequality. Each problem is broken down step-by-step, with solutions provided where applicable. Overall, clear understanding of arithmetic, sequences, and inequalities is demonstrated.
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