/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 18 (1) Without using your calculato... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 18

(1) Without using your calculator evaluate (a) \(\frac{4}{5} \times \frac{25}{28}\) (b) \(\frac{2}{7} \times \frac{14}{25} \times \frac{30}{48}\) (c) \(\frac{9}{16} \div \frac{3}{8}\) (d) \(\frac{2}{5} \times \frac{1}{12} \div \frac{8}{25}\) (e) \(\frac{10}{13}-\frac{2}{13}\) (f) \(\frac{5}{9}+\frac{2}{3}\) (g) \(2 \frac{3}{5}+1 \frac{3}{7}\) (h) \(5 \frac{9}{10}-\frac{1}{2}+1 \frac{2}{5}\) (i) \(3 \frac{3}{4} \times 1 \frac{3}{5}\) (j) \(\frac{3}{5} \times\left(\frac{2}{3}+\frac{1}{2}\right)\) (k) \(\frac{5}{6} \times\left(2 \frac{1}{3}-1 \frac{2}{5}\right)\) (I) \(\left(3 \frac{1}{3} \div 2 \frac{1}{6}\right) \div \frac{5}{13}\) (2) Confirm your answer to part (1) using a calculator.

Short Answer

Expert verified
a) \( \frac{5}{7}, \) b) \( \frac{1}{10} \) c) \( \frac{3}{2} \) d) \( \frac{5}{48} \) e) \(\frac{8}{13} \) f) \(1 \frac{2}{9} \) g) \frac{1}{35} \h) g)\ (\ = 6\) i\(is = \frac{7}{10}) j,\frac{12}{15} = (\frac{2}{46:})

Step by step solution

01

Evaluate \(\frac{4}{5} \times \frac{25}{28}\)

Multiply the numerators: \(4 \times 25 = 100\). Multiply the denominators: \(5 \times 28 = 140\). The result is \(\frac{100}{140} = \frac{5}{7}\).
02

Evaluate \(\frac{2}{7} \times \frac{14}{25} \times \frac{30}{48}\)

Multiply the numerators: \(2 \times 14 \times 30 = 840\). Multiply the denominators: \(7 \times 25 \times 48 = 8400\). Simplify: \(\frac{840}{8400} = \frac{1}{10}\).
03

Evaluate \(\frac{9}{16} \div \frac{3}{8}\)

Division of fractions is multiplication by the reciprocal. So, \(\frac{9}{16} \times \frac{8}{3} = \frac{9 \times 8}{16 \times 3} = \frac{72}{48} = \frac{3}{2}\).
04

Evaluate \(\frac{2}{5} \times \frac{1}{12} \div \frac{8}{25}\)

First multiply \(\frac{2}{5} \times \frac{1}{12} = \frac{2}{60} = \frac{1}{30}\). Then divide: \(\frac{1}{30} \times \frac{25}{8} = \frac{25}{240} = \frac{25}{240} = \frac{5}{48}\).
05

Evaluate \(\frac{10}{13} - \frac{2}{13}\)

Since the fractions have a common denominator: \(\frac{10}{13} - \frac{2}{13} = \frac{8}{13} \).
06

Evaluate \(\frac{5}{9} + \frac{2}{3}\)

Convert \(\frac{2}{3}\) to \(\frac{6}{9}\) so they have a common denominator: \(\frac{5}{9} + \frac{6}{9} = \frac{11}{9} = 1 \frac{2}{9}\).
07

Evaluate \(2 \frac{3}{5} + 1 \frac{3}{7}\)

Convert to improper fractions: \(2 \frac{3}{5} = \frac{13}{5}\) and \(1 \frac{3}{7} = \frac{10}{7}\). Find a common denominator: \(13 \times 7 = 91\) and \(5 \times 10 = 50\). Adding: \( \frac{91}{35} + \frac{50}{35} = \frac{141}{35} = 4 \frac{1}{35}\).
08

Evaluate \(5 \frac{9}{10} - \frac{1}{2} + 1 \frac{2}{5}\)

Convert to improper fractions and find common denominators: \(5 \frac{9}{10} = \frac{59}{10}\), \( \frac{1}{2} = \frac{5}{10} \), and \(1 \frac{2}{5} = \frac{7}{5} = \frac{14}{10}\). Subtraction and addition: \(\frac{59}{10} - \frac{5}{10} = \frac{54}{10} = \frac{27}{5}\), and \(\frac{27}{5} + \frac{14}{10} = 5 \frac{7}{10} \).
09

Evaluate \(3 \frac{3}{4} \times 1 \frac{3}{5}\)

Convert to improper fractions: \(3 \frac{3}{4} = \frac{15}{4}\) and \(1 \frac{3}{5} = \frac{8}{5}\). Multiply: \( \frac{15}{4} \times \frac{8}{5} = \frac{120}{20} = 6 \).
10

Evaluate \(\frac{3}{5} \times (\frac{2}{3} + \frac{1}{2})\)

First Add: \(\frac{2}{3} + \frac{1}{2} = \frac{4}{6} + \frac{3}{6} = \frac{7}{6}\). Then multiply: \(\frac{3}{5} \times \frac{7}{6} = \frac{21}{30} = \frac{7}{10} \).
11

Evaluate \(\frac{5}{6} \times (2 \frac{1}{3} - 1 \frac{2}{5})\)

First convert and subtract: \(2 \frac{1}{3} = \frac{7}{3}\) and \(1 \frac{2}{5} = \frac{7}{5} = \frac{21}{15}\). Subtract \( \frac{35}{15} - \frac{21}{15} = \frac{14}{15}\). Multiply: \( \frac{5}{6} \times \frac{14}{15} = \frac{70}{90} = \frac{7}{9}\).
12

Evaluate \(\frac{5}{6} \times (2 \frac{1}{3} - 1 \frac{2}{5})\)

First convert to improper fractions and divide \( \frac{10}{3}梅\frac{7}{6} \). Multiplying by the reciprocal: \frac{10}{3}脳\frac{6}{7}锛漒frac{20}{7}\
13

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Multiplication of Fractions
Multiplication of fractions is straightforward and involves two simple steps. First, multiply the numerators (the top numbers) of the fractions together. Then, multiply the denominators (the bottom numbers) together. Finally, simplify the resulting fraction if possible. Here鈥檚 an example:
To multiply \(\frac{4}{5}\) and \(\frac{3}{7}\), you multiply:
\[ \frac{4 \times 3}{5 \times 7} = \frac{12}{35} \]
This gives the final simplified fraction. Remember to always check if you can simplify further by finding the greatest common divisor (GCD) of the numerator and the denominator.
Division of Fractions
Dividing fractions can seem tricky, but it鈥檚 quite manageable when you understand the concept of multiplying by the reciprocal. The reciprocal of a fraction is created by swapping its numerator and denominator. To divide fractions, you simply multiply by the reciprocal of the second fraction. Let鈥檚 dive into an example:
To divide \(\frac{9}{16} \text{by} \ \frac{3}{8}\), you find the reciprocal of \(\frac{3}{8}\), which is \(\frac{8}{3}\) and then multiply:
\[ \frac{9}{16} \times \frac{8}{3} = \frac{9 \times 8}{16 \times 3} = \frac{72}{48} = \frac{3}{2} \]
Remember to simplify your fraction at the end.
Addition and Subtraction of Fractions
Adding and subtracting fractions requires a common denominator. The common denominator is a multiple of both denominators involved. Once the denominators match, you can simply add or subtract the numerators. Here is an example of adding fractions:
\(\frac{1}{4} + \frac{1}{6}\):
Find the least common denominator (LCD), which is 12 in this case. Convert each fraction:
\[ \frac{1}{4} = \frac{3}{12} \] and \[ \frac{1}{6} = \frac{2}{12} \]
Now add the new fractions: \[ \frac{3}{12} + \frac{2}{12} = \frac{5}{12} \]
For subtraction, the steps are the same, except you subtract the numerators instead of adding them.
Improper Fractions
An improper fraction is a fraction where the numerator (top number) is greater than or equal to the denominator (bottom number). This means the fraction is equal to or greater than one. To work with improper fractions, you often convert them to mixed numbers and vice versa.
Conversion to a mixed number involves dividing the numerator by the denominator to get a whole number and a remainder. For instance, converting \(\frac{9}{4}\) to a mixed number:
Divide 9 by 4, which equals 2 with a remainder of 1. Therefore, \(\frac{9}{4} = 2\frac{1}{4}\).
Conversely, converting a mixed number back to an improper fraction involves multiplying the whole number by the denominator and adding the numerator. For example, converting \(\frac{2\frac{1}{3}}\) to an improper fraction:
\[ 2 \times 3 + 1 = 7 \] which gives \[ \frac{7}{3} \].
Understanding these conversions is critical for solving many fraction problems accurately.

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Most popular questions from this chapter

Solve the following equations: (a) \(5(2 x+1)=3(x-2)\) (b) \(5(x+2)+4(2 x-3)=11\) (c) \(5(1-x)=4(10+x)\) (d) \(3(3-2 x)-7(1-x)=10\) (e) \(9-5(2 x-1)=6\) (f) \(\frac{3}{2 x+1}=2\) (g) \(\frac{2}{x-1}=\frac{3}{5 x+4}\) (h) \(\frac{x}{2}+3=7\) (i) \(5-\frac{x}{3}=2\) (j) \(\frac{5(x-3)}{2}=\frac{2(x-1)}{5}\) (k) \(\sqrt{(2 x-5)=3}\) (I) \((x+3)(x-1)=(x+4)(x-3)\) \((\mathrm{m})(x+2)^{2}+(2 x-1)^{2}=5 x(x+1)\) (n) \(\frac{2 x+7}{3}=\frac{x-4}{6}+\frac{1}{2}\) (o) \(\sqrt{\frac{45}{2 x-1}}=3\) (p) \(\frac{4}{x}-\frac{3}{4}=\frac{1}{4 x}\)

If the consumption function is given by $$ C=0.7 Y+40 $$ state the values of (a) autonomous consumption (b) marginal propensity to consume Transpose this formula to express \(Y\) in terms of \(C\) and hence find the value of \(Y\) when \(C=110\).

Sketch the following lines on the same diagram: $$ 2 x-3 y=6, \quad 4 x-6 y=18, \quad x-\frac{3}{2} y=3 $$ Hence comment on the nature of the solutions of the following systems of equations: (a) \(2 x-3 y=6\) (b) \(4 x-6 y=18\) \(x-\frac{3}{2} y=3\) \(x-\frac{3}{2} y=3\)

The demand and supply functions for two interdependent commodities are given by $$ \begin{aligned} &Q_{\mathrm{D}_{1}}=40-5 P_{1}-P_{2} \\ &Q_{\mathrm{D}_{2}}=50-2 P_{1}-4 P_{2} \\ &Q_{\mathrm{s}_{1}}=-3+4 P_{1} \\ &Q_{\mathrm{s}_{2}}=-7+3 P_{2} \end{aligned} $$ where \(Q_{D_{i}}, Q_{s,}\) and \(P_{i}\) denote the quantity demanded, quantity supplied and price of good \(i\) respectively. Determine the equilibrium price and quantity for this two-commodity model. Are these goods substitutable or complementary?

(Excel) The supply and demand functions of a good are given by $$ \begin{aligned} &P=-Q_{\mathrm{D}}+240 \\ &P=60+2 Q_{\mathrm{s}} \end{aligned} $$ where \(P, Q_{D}\) and \(Q_{5}\) denote price, quantity demanded and quantity supplied, respectively. Sketch graphs of both functions on the same diagram, on the range \(0 \leq Q \leq 80\) and hence find the equilibrium price. The government now imposes a fixed tax, \(\$ 60\), on each good. Draw the new supply equation on the same diagram and hence find the new equilibrium price. What fraction of the \(\$ 60\) tax is paid by the consumer? Consider replacing the demand function by the more general equation $$ P=-k Q_{\mathrm{D}}+240 $$ By repeating the calculations above, find the fraction of the tax paid by the consumer for the case when \(k\) is (a) 2 (b) 3 (c) 4 State the connection between this fraction and the value of \(k\). Use this connection to predict how much tax is paid by the consumer when \(k=6\).
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