# Rates and Ratios

## Table of Contents

## Like and Unlike Measurements

- This unit deals with the idea of “like” and “unlike” units, so it’s important to understand what these are.
- “Like” measurements are measurements which measure the same thing (e.g. time, distance, mass)
- For example, grams, kilograms, and tonnes are all like units, because they all measure mass.
- Seconds, minutes, and hours are like units, because they all measure time.
- Gallons and Litres are also like units, because they both measure volume.

- Unlike measurements have 2 or more different units involved.
- Speed, for example, is an unlike measurement, because it’s measured in meters (unit of distance) per second (unit of time).
- Since time can’t be converted into a distance, and distance can’t be converted into a time, we say that speed is an “unlike measurement”.
- However, comparing the speed of 2 objects is a like measurement, because both measurements will be of speed.

## One way to figure out whether 2 measurements are like or unlike is to divide one set’s units by the other’s (skip this if you don’t like algebra).

- For example, say you have a table of weights versus a table of masses, and you want to figure out if they are like or unlike:
- The unit for weight is the Newton, which is equal to kilograms times meters per second squared $\left(1N=1kg\cdot m/s^2\right)$.
- The unit for mass is the kilogram.
- Eliminating Kilograms from Newtons looks like $\frac{\cancel{kg}\cdot m/s^2}{\cancel{kg}}=m/s^2$.
- Since all units weren’t eliminated, kilograms and newtons are unlike.

- If we try grams and tonnes, however:
- $1 tonne = 1,000 kg = 1,000,000\color{orange}{g}$
- $\frac{\cancel{g}}{\cancel{g}}=1,$ therefore tonnes and grams are equivalent units.

## Rates

- A rate is a comparison of 2 unlike quantities.
- Speed is an example of a rate, as it is a comparison of distance (meters or kilometers) with time (seconds or hours)

### Heart Rate

- A person’s heart rate is measured by the amount of time it beats each minute.
- Naturally, the unit is Beats per Minute (BPM)
- The resting heart rate for most people is between 60 and 80BPM
- The maximum heart rate (MHR) is different based on age, but can be estimated as 220 minus the person’s age in years.
- The Exercise Target Heart Rate is the heart rate you typically aim for when doing exercise, and is between 65% and 85% of that person’s MHR.

### Energy Rate

- Energy is the rate of power per unit of time, measured inkilowatt hours $(kWh)$.
- Running cost of the appliance is calculated by multiplying the energy consumption by the electricity price

$$\text{Cost}=\frac{\text{Energy Consumption }(kWh)}{\text{Electricity Price }($ /kWh)}$$

### Fuel Consumption Rate

Fuel consumption is measured in Litres per Kilometer $\text{(L/km)}$

$$\text{Fuel consumption (L/km)} = \frac{\text{Amount of Fuel (L)}\times100}{\text{Distance Travelled (km)}}$$

## Ratios

- A ratio is used to compare amounts of the same units in a definite order.
- Equivalent ratios are obtained by multiplying or dividing by the same number (e.g. $\frac{2}{3}=\frac{4}{6}=\frac{20}{30}$)

### Dividing a Quantity in a Given Ratio

- Find the total number of parts by adding each amount in the ratio.
- Divide the quantity by the total number of parts to find one part.
- Multiply each amount of the ratio by the result in step 2.

### Scale Drawings

The scale of a drawing is the ratio between the drawing length and the actual length it represents.

$$\text{Scale}=\text{drawing length}:\text{actual length}$$

Scale can be expressed with or without units.

- For example, if 1cm on a map is 1m in real life, it can be written as $1cm:1m$ or $1:100$

#### Perimeter and Area

- This principle can be extended to perimeter and area.
- To find the perimeter of an object on a map, find the perimeter on the map, and multiply by the scale.
- To estimate the area of land, divide it into square grids and count the number of squares.
- To calculate the area of land, use the appropriate formula.

Shape | Area Formula |
---|---|

Triangle | $A=\frac{1}{2}bh$ or $A=\frac{1}{2}ab\times\cos(C)$ |

Rectangle | $A=lb$ |

Square | $A=l^2$ |

Parallelogram | $A=bh$ |

Trapezium | $A=\frac{1}{2}(a+b)h$ |

Rhombus | $A=\frac{1}{2}xy$ |

### Volume of Water and the Trapezoidal Rule

- Volume of a body of water of constant depth and a trapezoidal area, or volume of rainfall falling on a trapezoidal area, can be calculated using the trapezoidal rule:

$$A\approx\frac{w}{2}(df+dl)$$

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