Apply percentage changes and compare percentage errors

What is twelve percent of two hundred?

Keep the original and add the change

The original amount is the base represented by one hundred percent. A percentage increase adds a percentage of that base

The new amount is the original whole plus fifteen hundredths of it

Finding the change first and using a single multiplier give the same exact result

Construct a multiplier for a fractional percentage

A quantity has a percentage increase of 12.5%

Construct the single multiplier as a decimal or whole number for the stated percentage change

The original amount represents one whole, or one hundred percent

An increase adds the extra fraction to the original whole

Include the original whole in the multiplier

Keep only the fraction left after a decrease

After a twenty percent decrease, eighty percent of the original amount remains

Multiplying by zero point two would give the amount removed, not the amount remaining

A decrease of a non-negative quantity cannot exceed one hundred percent without making the result negative

Subtract the percentage fraction from one

A quantity has a percentage decrease of 25%

Construct the single multiplier as a decimal or whole number for the stated percentage change

The original amount represents one whole, or one hundred percent

A decrease keeps the fraction left after subtracting the stated percentage

Construct the fraction remaining

Show the change and the new amount separately

The original amount is $80. Apply a 15% increase

Find the exact size of the change only, before combining it with the original amount

Divide the percentage by one hundred, then multiply the original amount

Find the amount after the increase

Give the exact amount as a decimal or whole number

First find the exact size of the change from the original amount

Add that change to the original amount

Alternatively, combine the original whole and the percentage into one multiplier

The original amount represents one whole, or one hundred percent

An increase adds the extra fraction to the original whole

The result is exact in dollars

Construct both stages of an increase

Subtract a positive change size from the original

The original amount is $120. Apply a 15% decrease

Find the exact size of the change only, before combining it with the original amount

Divide the percentage by one hundred, then multiply the original amount

Find the amount after the decrease

Give the exact amount as a decimal or whole number

First find the exact size of the change from the original amount

Subtract that change from the original amount

Alternatively, combine the original whole and the percentage into one multiplier

The original amount represents one whole, or one hundred percent

A decrease keeps the fraction left after subtracting the stated percentage

The result is exact in dollars

Do not confuse the removed amount with the final amount

Translate the context into an increase or decrease

A discount reduces the stated original price; a mark-up adds a percentage of the stated cost or base

Read which amount supplies the base. A mark-up of a cost is not the same percentage calculation as a profit margin based on selling price

For a tax added to a tax-exclusive price, increase that price by the stated tax rate. The problem must tell you the applicable rate and base

For a population model, an exact percentage calculation may give an integer count. If it does not, the context needs an explicit rounding or interpretation rule

Find a price after a stated discount

Use the stated cost as the mark-up base

An item costs a seller $50. The seller adds a mark-up of 30% of that cost. Find the selling price before any tax

First find the exact size of the change from the original amount

Add that change to the original amount

Alternatively, combine the original whole and the percentage into one multiplier

The original amount represents one whole, or one hundred percent

An increase adds the extra fraction to the original whole

The result is exact in dollars

Add a supplied tax rate to the exclusive price

Calculate a count in a stated growth model

Check large increases and the zero boundary

An increase can exceed one hundred percent: the added amount can be larger than the original

A one hundred percent decrease removes the whole original amount, leaving zero

A zero percent change leaves the original amount unchanged

Allow an added amount larger than the original

Apply the full-removal boundary

Check a zero percent change

The original amount is 80 units. Apply a 0% increase

Find the amount after the increase

Give the exact amount as a decimal or whole number

First find the exact size of the change from the original amount

Add that change to the original amount

Alternatively, combine the original whole and the percentage into one multiplier

The original amount represents one whole, or one hundred percent

An increase adds the extra fraction to the original whole

The result is exact in units

Carry exact values until the requested final rounding

Use mental or written methods for convenient values. A calculator or spreadsheet helps with less convenient decimal inputs, but the same base and multiplier still apply

If the question requests a price to the nearest cent, round this final result to two decimal places

Do not round the percentage change before adding it to the original. Early rounding can change the final answer

In these rounding tasks, the requested number of decimal places is part of the written answer: a whole-dollar result may need two trailing zeros

Round the final amount, not its intermediate change

The original amount is $19.95. Apply a 7.5% increase

Find the amount after the increase

Round only the final amount to 2 decimal places

First find the exact size of the change from the original amount

Add that change to the original amount

Alternatively, combine the original whole and the percentage into one multiplier

The original amount represents one whole, or one hundred percent

An increase adds the extra fraction to the original whole

Keep every intermediate value exact and round only this final result

Write exactly 2 decimal places in dollars

Retain all intermediate decimal places

Calculate change relative to the original amount

When original and final amounts are given, subtract original from final to obtain the signed change

The original amount must be positive in these tasks. A positive result is an increase; a negative result is a decrease

If asked for the size of the percentage decrease, report the non-negative magnitude twenty percent instead of the signed negative result

Divide by the original amount, not the final amount. Both amounts must use compatible units

Recover a percentage increase from two amounts

Original amount: $80

Final amount: $100

Find the signed percentage change; use a negative sign for a decrease

Give an exact number of percent; use a fraction in simplest form if its decimal expansion recurs

Use the original amount, not the final amount, as the comparison base

Divide the signed change by the original amount and multiply by one hundred

The signed result is positive, showing an increase

Keep this exact value rather than copying a rounded display

Retain the sign when the question asks for signed change

Report the requested decrease magnitude

Original amount: $80

Final amount: $68

Find the nonnegative percentage decrease

Give an exact number of percent; use a fraction in simplest form if its decimal expansion recurs

Use the original amount, not the final amount, as the comparison base

Divide the signed change by the original amount and multiply by one hundred

The signed result is negative, showing a decrease

The question asks for the size of the decrease, so report its nonnegative magnitude

Keep this exact value rather than copying a rounded display

A percentage increase can exceed one hundred

An exact percentage can be fractional or recurring

The exact number before the percent sign is ten thirds, which is recurring in decimal form

When exactness is requested, enter the simplified fraction rather than replacing it by three point three three

A request for two decimal places would instead require the stated rounded result. Exact and rounded tasks have different answer requirements

Preserve an exact recurring percentage

Original amount: 30 units

Final amount: 31 units

Find the signed percentage change; use a negative sign for a decrease

Give an exact number of percent; use a fraction in simplest form if its decimal expansion recurs

Use the original amount, not the final amount, as the comparison base

Divide the signed change by the original amount and multiply by one hundred

The signed result is positive, showing an increase

Keep this exact value rather than copying a rounded display

Construct an exact signed fraction of a percent

Round a signed percentage only at the end

Original amount: $120

Final amount: $100

Find the signed percentage change; use a negative sign for a decrease

Round only the final percentage to 2 decimal places

Use the original amount, not the final amount, as the comparison base

Divide the signed change by the original amount and multiply by one hundred

The signed result is negative, showing a decrease

Write exactly 2 decimal places and retain the requested sign convention

Use the original base before final rounding

A price falls from $100 to $80, then rises from $80 to $100. Which description is correct?

Error compares a reading or estimate with a stated reference

Treat the stated positive reference as exact for each error task. The reading or estimate is the value being checked

Absolute error is the non-negative size of the difference and retains the quantity unit

Relative error divides that size by the positive reference, so compatible units cancel

Percentage error is one hundred times that ratio

Unlike signed percentage change, percentage error is non-negative. A reading below the reference does not give a negative error

Put the quantities in the same unit before subtracting

A short metric bridge makes the comparison base explicit

A reference of one point two metres and a reading of one hundred and eighteen centimetres must not be subtracted as 1.2 minus 118

Conversion

\(\displaystyle 1\text{ m}=100\text{ cm}\)

A reference length is 1.5 m and a reading is 147 cm. Enter the reference length in centimetres before calculating any error

Multiply the metre measure by one hundred

Find only the non-negative discrepancy

the stated reference is 200 cm and the reading is 194 cm. Treat the positive reference as exact

Calculate the absolute error in cm

Keep the answer exact; do not round

Use the stated positive reference as exact and keep both values in the same unit

Find the non-negative size of the difference

The reading is below the reference, but the error magnitude is positive

Keep the absolute error unit

Divide the discrepancy by the reference

the stated reference is 200 cm and the reading is 194 cm. Treat the positive reference as exact

Calculate the relative error as a ratio without a percent sign

Keep the answer exact; do not round

Use the stated positive reference as exact and keep both values in the same unit

Find the non-negative size of the difference

The reading is below the reference, but the error magnitude is positive

Divide the absolute error by the reference, not by the reading. The measurement units cancel

Return the ratio without multiplying by one hundred

Express the relative error as a percentage

the stated reference is 200 cm and the reading is 194 cm. Treat the positive reference as exact

Calculate the percentage error

Keep the answer exact; do not round

Use the stated positive reference as exact and keep both values in the same unit

Find the non-negative size of the difference

The reading is below the reference, but the error magnitude is positive

Divide the absolute error by the reference, not by the reading. The measurement units cancel

Multiply the relative error by one hundred to obtain the percentage number

Use the reference rather than the reading as denominator

Check equality, zero readings and large errors

Equal reading and reference give zero absolute, relative and percentage error

A zero reading is allowed when the reference is positive

Percentage error can exceed one hundred percent if the discrepancy exceeds the reference

A zero reference is different: this percentage-error formula would divide by zero and is not defined

The interpretation uses the stated reference as exact; it does not account for uncertainty in that reference

Check a matching reading and reference

the stated reference is 50 cm and the reading is 50 cm. Treat the positive reference as exact

Calculate the percentage error

Keep the answer exact; do not round

Use the stated positive reference as exact and keep both values in the same unit

Find the non-negative size of the difference

The reading equals the reference, so the error is zero

Divide the absolute error by the reference, not by the reading. The measurement units cancel

Multiply the relative error by one hundred to obtain the percentage number

A zero reading does not mean zero error

Compare a discrepancy larger than the reference

A proposed percentage-error calculation has reference zero and reading five. What should you conclude?

Keep the error ratio exact until the final percentage

the stated reference is 75 cm and the reading is 73 cm. Treat the positive reference as exact

Calculate the percentage error

Round only the final answer to 2 decimal places

Use the stated positive reference as exact and keep both values in the same unit

Find the non-negative size of the difference

The reading is below the reference, but the error magnitude is positive

Divide the absolute error by the reference, not by the reading. The measurement units cancel

Multiply the relative error by one hundred to obtain the percentage number

Keep the calculation exact until this final step, then round to 2 decimal places

Round the final number before the percent sign

The larger absolute discrepancy need not be the larger relative error

Compare each discrepancy with its own reference, not with a shared denominator chosen arbitrarily

Reading B has a larger absolute discrepancy of ten but a smaller relative error of one percent

When comparing rounded percentages, retain the exact ratios to decide the winner. Two displayed rounded percentages may be equal even though the exact errors differ

Calculate both errors before comparing their relative sizes

Reading A: the stated reference is 100 units and the reading is 104 units. Treat the positive reference as exact

Reading B: the stated reference is 1000 units and the reading is 1010 units. Treat the positive reference as exact

For reading A, calculate the percentage error

Keep the answer exact; do not round

Use the stated positive reference as exact and keep both values in the same unit

Find the non-negative size of the difference

The reading is above the reference

Divide the absolute error by the reference, not by the reading. The measurement units cancel

Multiply the relative error by one hundred to obtain the percentage number

For reading B, calculate the percentage error

Keep the answer exact; do not round

Use the stated positive reference as exact and keep both values in the same unit

Find the non-negative size of the difference

The reading is above the reference

Divide the absolute error by the reference, not by the reading. The measurement units cancel

Multiply the relative error by one hundred to obtain the percentage number

Identify the reading with the smaller relative error, or state that the errors are equal. Use exact ratios before rounding

Reading A

Use the stated positive reference as exact and keep both values in the same unit

Find the non-negative size of the difference

The reading is above the reference

Divide the absolute error by the reference, not by the reading. The measurement units cancel

Reading B

Use the stated positive reference as exact and keep both values in the same unit

Find the non-negative size of the difference

The reading is above the reference

Divide the absolute error by the reference, not by the reading. The measurement units cancel

Use exact relative errors even if the percentage answers have been rounded

Reading B has the smaller relative error

Use the positive reference belonging to each reading

Recognise equal relative errors at different scales

Do not decide exact equality from identical rounded percentages

R is a positive base and D is final minus base. In a change context use R as the original amount; in an error context use R as an exact reference and the reading as the reference plus the signed difference. Compare negative and positive D. The percentage readouts are rounded to two decimal places

What changes when D moves from negative ten to positive ten with R held at one hundred?

Compare exact reasoning with a digital calculation

This investigation is not automatically marked. Choose an original price and a stated discount or mark-up, and calculate the change and final price by finding the change first and by using a single multiplier

Check with a calculator or spreadsheet. If a price must be rounded to cents, show the exact intermediate result and the final rounded result separately

Create two reading-reference pairs with different positive references. Compute both percentage errors and explain why comparing only their absolute discrepancies can be misleading

Use units consistently and state whether the reference is being treated as exact

A hypothetical tax task must supply its rate and applicability; do not infer current tax rules from the examples here

State the base, the requested quantity and the precision

The change amount alone is not the new amount. A discount subtracts a percentage of its stated original base

A requested increase or decrease magnitude is non-negative; a signed percentage change retains its direction

The reference is positive and stated as exact here. Use compatible units and distinguish absolute, relative and percentage error

Compare exact relative errors before rounding. A larger absolute discrepancy can be a smaller percentage error

Keep exact fractions or decimals until the requested final rounding, and write the required precision