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Chapter 3: Problem 11

Write down the scale factors corresponding to (a) an increase of \(19 \%\) (b) an increase of \(250 \%\) (c) a decrease of \(2 \%\) (d) a decrease of \(43 \%\)

Short Answer

Expert verified
1.19, 3.50, 0.98, 0.57

Step by step solution

01

Understanding Percentage Increase

When increasing a value by a percentage, convert the percentage to a decimal and add 1. This represents the new scale factor.
02

Calculate Scale Factor for 19% Increase

Convert 19% to a decimal: \(19\% = \frac{19}{100} = 0.19\).Add 1 to the decimal: \(1 + 0.19 = 1.19\).The scale factor for a 19% increase is 1.19.
03

Calculate Scale Factor for 250% Increase

Convert 250% to a decimal: \(250\% = \frac{250}{100} = 2.50\).Add 1 to the decimal: \(1 + 2.50 = 3.50\).The scale factor for a 250% increase is 3.50.
04

Understanding Percentage Decrease

When decreasing a value by a percentage, convert the percentage to a decimal and subtract it from 1. This represents the new scale factor.
05

Calculate Scale Factor for 2% Decrease

Convert 2% to a decimal: \(2\% = \frac{2}{100} = 0.02\).Subtract the decimal from 1: \(1 - 0.02 = 0.98\).The scale factor for a 2% decrease is 0.98.
06

Calculate Scale Factor for 43% Decrease

Convert 43% to a decimal: \(43\% = \frac{43}{100} = 0.43\).Subtract the decimal from 1: \(1 - 0.43 = 0.57\).The scale factor for a 43% decrease is 0.57.

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Key Concepts

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

percentage increase
In mathematics, a percentage increase is used to show how much a quantity has grown in relation to its original value. To find the scale factor for a percentage increase, you first need to convert the percentage to a decimal. For example, if you have a 19% increase, you convert 19% to its decimal form by dividing by 100: \( 19\% = \frac{19}{100} = 0.19 \). Next, you add 1 to the decimal value to get the new scale factor: \( 1 + 0.19 = 1.19 \). This means that if the original value was multiplied by 1.19, it reflects a 19% increase.
percentage decrease
A percentage decrease indicates how much a value has reduced relative to its original amount. To find the scale factor for a percentage decrease, start by converting the percentage to a decimal. For instance, a decrease of 2% is written as: \( 2\% = \frac{2}{100} = 0.02 \). Now, instead of adding this to 1, you subtract it from 1: \( 1 - 0.02 = 0.98 \). This means the scale factor for reducing a value by 2% is 0.98. The original value is thus multiplied by 0.98 to reflect a 2% decrease.
mathematical conversion
Mathematical conversion involves changing one type of number into another. This often includes converting percentages to decimals and vice versa. Converting percentages to decimals is straightforward: divide the percentage by 100. For example: \( 250\% = \frac{250}{100} = 2.50 \). This can easily be understood as 'moving the decimal point two places to the left'. To convert in the opposite direction, multiplying the decimal by 100 shifts the decimal point two places to the right, turning 2.50 back into 250%. This basic conversion skill is essential for understanding and manipulating mathematical expressions involving percentages.
decimal representation
Decimal representation is the way of expressing numbers using a base-10 numeral system. It is often used to represent percentages. In problems involving percentage increases or decreases, converting into decimal form simplifies calculations. Consider converting 43% to a decimal: \( 43\% = \frac{43}{100} = 0.43 \). When this number is used in calculations, it becomes much easier to manage. The consistency and simplicity of decimals make them valuable in many areas of mathematics. They provide a straightforward way to multiply and divide by powers of ten, making calculations more intuitive.

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Most popular questions from this chapter

If a principal, \(P\), is invested at \(r \%\) interest compounded annually then its future value, \(S\), after \(n\) years is given by $$ S=P\left(1+\frac{r}{100}\right)^{n} $$ (a) Use this formula to show that if an interest rate of \(r \%\) is compounded \(k\) times a year then after \(t\) years $$ S=P\left(1+\frac{r}{100 k}\right)^{k t} $$ (b) Show that if \(m=100 k / r\) then the formula in part (1) can be written as $$ S=P\left(\left(1+\frac{1}{m}\right)^{m}\right)^{r t / 100} $$ (c) Use the definition $$ \mathrm{e}=\lim _{m \rightarrow \infty}\left(1+\frac{1}{m}\right)^{m} $$ to deduce that if the interest is compounded with ever-increasing frequency (that is, continuously) then $$ S=P \mathrm{e}^{r / 100} $$

An investment project requires an initial outlay of \(\$ 8000\) and will produce a return of \(\$ 17000\) at the end of 5 years. Use the (a) net present value (b) internal rate of return methods to decide whether this is worthwhile if the capital could be invested elsewhere at \(15 \%\) compounded annually.

A regular saving of \(\$ 500\) is made into a sinking fund at the start of each year for 10 years. Determine the value of the fund at the end of the tenth year on the assumption that the rate of interest is (a) \(11 \%\) compounded annually (b) \(10 \%\) compounded continuously

Determine the APR if the nominal rate is \(7 \%\) compounded continuously.

Determine the present value of \(\$ 7000\) in 2 years' time if the discount rate is \(8 \%\) compounded (a) quarterly (b) continuously
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