Everyday sums

Fractions and ratios

Converting fractions to percentages, fraction arithmetic, the rule of three and scale.

A fraction, a percentage and a decimal are three ways of writing the same value — 3/8 is 0.375 and 37.5 %. A proportion carries a known relationship over to another quantity, and that is enough to scale recipes and mixtures, read map scales and split an amount between several people.

Calculators

Fraction → percentage

p = l / m · 100 %

Numerator

Denominator

Percentage37.50%

Decimal0.375
After reducing3/8
Per mille375 ‰

Percentage → fraction

u = p / 100

Percentage%

Fraction3/8

Decimal0.375
Per mille375 ‰
ppm375,000

Fraction arithmetic

a/b ⊕ c/d

Numerator 1

Denominator 1

Numerator 2

Denominator 2

Sum5/6

Difference1/2
Product1/9
Quotient4/1
Sum in decimal0.833333

Reducing a fraction

n / d ÷ GCD

Numerator

Denominator

After reducing2/3

GCD72
Mixed number0 + 2/3
Decimal0.666667
As a percentage66.67%

Proportion (rule of three)

a : b = c : x

If a

then b

then what for c

x28

Factor b / a4
Inverse proportion5.142857

Split in a ratio

part = whole · aᵢ / Σa

Whole

Part A

Part B

Part C

Part A600

Part B400
Part C200
Share A50.00%
One part200

Map scale

ground = line · S

Scale 1 : S

Line on the map (cm)

On the ground1,125 m

In kilometers1.125 km
1 cm on the map250 m
1 km on the ground4 cm

Tables and cheat sheets

Map scale in practice

how much ground a line on the map covers

Scale	1 cm on the map	1 km on the ground	Typical use
1 : 500	5 m	200 cm	plot plan
1 : 1,000	10 m	100 cm	development plan
1 : 5,000	50 m	20 cm	city plan
1 : 10,000	100 m	10 cm	hiking map
1 : 25,000	250 m	4 cm	topographic map
1 : 50,000	500 m	2 cm	topographic map
1 : 100,000	1 km	1 cm	road map
1 : 500,000	5 km	0.2 cm	regional map
1 : 1,000,000	10 km	0.1 cm	country map

Worked examples with full solutions

Splitting an amount in the ratio 3 : 2 : 1

$1,200 has to be split between three people in the ratio 3 : 2 : 1.

Add up the parts3 + 2 + 1 = 6 parts
Value of one part1200 / 6 = $200
Split3 · 200 = $600, 2 · 200 = $400, 1 · 200 = $200
Check the total600 + 400 + 200 = $1,200

Answer: $600, $400 and $200 — shares of 50 %, 33.3 % and 16.7 %.

A distance on a 1 : 25,000 map

A route measures 4.5 cm on a hiking map. How far is that on the ground?

1 cm on the map is 25,000 cm on the ground25,000 cm = 250 m
Convert the line4,5 · 250 m = 1 125 m
Convert to kilometers1,125 m = 1.125 km

Answer: The route is 1.125 km on the ground.