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This is Quantities, Units, Fractions and Ratios.

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This walkthrough uses a generic synthetic narrator, not a
recording of Craig Wright.

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Pause whenever you need time to calculate.

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The original lesson, written walkthrough and exercises
accompany this video.

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A number needs a meaning.

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Begin by separating the calculation from the interpretation.

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The calculation 12 times 2 gives 24, but it cannot tell us
whether that answer describes pounds, kilograms

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or hours.

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We need the question for that.

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Write the unit beside each starting number, and carry it
through to your answer.

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This small habit prevents many economic mistakes because
prices, wages, output and time are different kinds

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of quantities.

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You do not need to rush.

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For each example, first explain in ordinary words what the
answer is supposed to describe, then calculate

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it.

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Totals and rates answer different questions.

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Imagine being told that someone earns twelve pounds.

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Without a time period, we do not know much about their pay.

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Twelve pounds per hour gives a rate; forty-eight pounds for
four hours gives a total.

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Notice how the hours in the rate and the hours worked fit
together.

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If we divide the forty-eight-pound total by four hours, we
recover twelve pounds per hour.

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Write rates as a fraction if that helps: pounds divided by
hours.

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Keep the denominator visible, because pounds per hour and hours
per pound describe different questions.

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Convert units before combining quantities.

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Think of conversion as choosing a different-sized measuring
unit.

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A kilogram is a larger unit than a gram, so the same amount
needs fewer kilograms than grams.

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That is why we divide the gram figure by one thousand.

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Going back from kilograms to grams requires multiplication by
one thousand.

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A quick size check is useful: three hundred grams should be
less than one kilogram, so an answer of three

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hundred thousand kilograms cannot be right.

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Time needs special care because an hour contains sixty minutes,
rather than one hundred.

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Always write the conversion relationship first.

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Fractions describe equal parts.

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The denominator is not just another number to manipulate.

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It tells you what one part means.

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If a twenty-pound amount is split into four equal parts, each
part is five pounds.

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Taking three of those parts gives fifteen pounds.

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The parts must be equal for this interpretation to work.

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A fraction can also be larger than one: five quarters means
five pieces of size one quarter, or one whole

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and one quarter.

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When using a fraction in an economic question, identify the
whole explicitly.

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Three quarters of income and three quarters of spending may be
different amounts.

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Equivalent fractions and decimal notation.

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Picture six shaded pieces in a whole cut into eight equal
pieces.

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Pair the small pieces together and you have three shaded pairs
out of four pairs.

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The shaded share has not changed, even though its name has.

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This is why simplifying requires the same operation on the
numerator and denominator.

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Dividing only the numerator would change the fraction.

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Decimal notation gives another useful name for the same value.

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For example, one third is approximately zero point three three
three, but rounding it to two decimal places

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introduces a small difference.

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Keep extra digits during a calculation and round at the end.

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Calculate with fractions carefully.

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Addition combines pieces of the same size, so halves and thirds
must first be renamed as sixths.

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Do not add denominators: doing that would change the size of
the pieces while you are counting them.

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Multiplication answers a different question, such as finding
two thirds of three fifths of a whole.

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Division asks how many of one quantity fit into another.

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For example, one half divided by one quarter equals one half
multiplied by four, which equals two.

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Two quarters fit inside one half.

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You cannot divide by zero because there is no meaningful finite
answer to that operation.

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Ratios compare; rates include units.

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Read a ratio in the stated order.

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Two to three for red to blue does not mean that two thirds of
all the items are red.

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The three counts blue parts, not total parts.

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Add the two and three to find five parts in the combined whole.

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A rate also compares quantities, but usually retains different
measurement units.

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If a purchase costs money and has a mass, dividing cost by mass
produces pounds per kilogram.

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That gives a common basis for comparison.

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Ratios alone show relative size; an actual total or another
known quantity is needed to recover absolute

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amounts.

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Worked example: plan a unit-price calculation.

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We start by naming both requested answers.

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The first must be in pounds per kilogram; the second must be in
pounds.

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The packet mass is given in grams, so it does not yet match the
kilogram denominator we want.

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Divide seven hundred and fifty by one thousand to obtain zero
point seven five kilograms.

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Now divide the packet price by that mass.

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Do not multiply the price by zero point seven five, because
that would not find a price for one kilogram.

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Pause before calculating and predict that a full kilogram
should cost more than this three-quarter-kilogram

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packet.

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Worked example: calculate and verify the cost.

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Multiplying both numbers in the division by one hundred removes
the decimal places without changing the

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quotient.

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One hundred and eighty divided by seventy-five is two point
four, giving a rate of two pounds forty per

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kilogram.

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We then multiply that rate by four and a half kilograms.

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Splitting four and a half into four plus one half makes the
arithmetic visible.

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The independent pack-count check reaches the same total.

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This example assumes the same price rate applies throughout,
with no discounts or extra charges.

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Six whole packs are available here, so the requested mass is
also achievable using this packaging.

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Worked example: divide a budget in a ratio.

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A ratio gives instructions for relative shares, rather than
fixed pound amounts.

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Our first task is to discover what one ratio part is worth.

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Add all the parts because the complete budget must cover all
three categories.

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Nine equal parts share three hundred and sixty pounds, so each
part is forty pounds.

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At this stage, do not assign forty pounds to every category.

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Rent is entitled to five of these equal parts, while travel
gets just one.

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Writing the category names directly above the ratio numbers is
a simple way to preserve the correct ordering

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throughout the calculation.

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Worked example: allocate and check the whole.

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We have now translated relative shares into actual money.

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Two checks are helpful.

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First, add the three allocations to confirm that the whole
budget has been assigned exactly once.

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Second, compare the allocations with the original ratio.

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Two hundred to one hundred and twenty to forty simplifies,
after dividing each amount by forty, to five to

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three to one.

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The food share also illustrates the difference between a ratio
comparison and a fraction of the whole.

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Food to rent is three to five, but food as a share of the
complete budget is three out of nine parts.

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Direct proportion: scale the same relationship.

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A unit-rate method often feels clearer than a memorised
cross-multiplication rule.

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Reduce the known bundle to one item, then build the new bundle.

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Four tickets to seven tickets is a multiplication by seven
quarters, so multiplying eighteen pounds by seven

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quarters would give the same result.

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But the method depends on a constant relationship.

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A booking fee, bulk discount or minimum charge would change the
structure.

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In economic questions, read the words before assuming
proportionality.

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If there is a fixed fee, total cost might still rise with
quantity, but it will not necessarily rise in

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exactly the same proportion.

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Use estimates and sensible precision.

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Calculators can produce many digits, but extra digits do not
automatically make an answer more meaningful.

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A purchase total expressed in pounds normally ends at pennies,
while a unit price might need extra decimal

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places for a fair comparison.

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Estimate before relying on the exact calculation.

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If ten items cost just under three pounds each, a result near
three hundred pounds is a warning.

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Also distinguish rounding from changing the underlying
assumptions.

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Rounding a price slightly is an approximation; assuming a
discount exists is a different model.

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Explain any assumption that materially affects what your answer
means.

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Your quantity-checking routine.

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You now have a routine that will carry into every later lesson.

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Before pressing calculator buttons, decide what the answer
should measure.

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During the calculation, keep units and category labels visible.

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Afterwards, check the total, reverse the operation, or estimate
the expected size.

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These checks are complementary: a tidy arithmetic answer can
still use the wrong whole or the wrong unit.

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In the exercises, show yourself the working on paper even when
the submission box asks only for a number.

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Enter numerical answers without currency symbols where the
field already supplies the unit, and use the

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precision requested by the question.