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A quick practice set of problems for education students to take in preparation for their numeracy test.

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An item, originally priced at $\\$\\var{original_price}$, is on sale with $\\var{discount_percent}\\%$ off. To determine the sale price:

Instead of determining $\\var{discount_percent}\\%$ of the original price and subtracting that from the original, we can simply find [[0]] $\\%$ of the original price.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

Sale price	$ = $	[[0]] $\\%$ of $\\$$ [[1]]
	$ = $	[[2]] $\\times$ [[1]]
	$ = $	[[3]]

We think of this as scaling/stretching/shrinking the original price by multiplying by [[2]]. This multiplication approach is often more useful than a subtraction approach.

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The original amount is a full $100\\%$ of the original. So if we remove $\\var{discount_percent}\\%$ from the $100\\%$ we have $100\\%-\\var{discount_percent}\\%=\\var{remainder_percent}\\%$ left.

To find the discounted price we can find $\\var{remainder_percent}\\%$ of $\\$\\var{original_price}$ by calculating $\\var{remainder_decimal}\\times \\var{original_price}=\\var{sale_price}$.

Therefore the sale price is $\\$\\var{sale_price}$.

A scaffolded question on increasing a price by a percentage

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An item, originally priced at $\\$\\var{original_price}$, has its price increased by $\\var{discount_percent}\\%$. To determine the new price:

Instead of determining $\\var{discount_percent}\\%$ of the original price and adding that to the original, we can simply find [[0]] $\\%$ of the original price.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

New price	$ = $	[[0]] $\\%$ of $\\$$ [[1]]
	$ = $	[[2]] $\\times$ [[1]]
	$ = $	[[3]]

We think of this as scaling/stretching the original price by multiplying by [[2]]. This multiplication approach is often more useful than the addition approach.

The original amount is a full $100\\%$ of the original. So if we add $\\var{discount_percent}\\%$ to the $100\\%$ we have $100\\%+\\var{discount_percent}\\%=\\var{remainder_percent}\\%$ of the original.

To find the new price we can find $\\var{remainder_percent}\\%$ of $\\$\\var{original_price}$ by calculating $\\var{remainder_decimal}\\times \\var{original_price}=\\var{sale_price}$.

Therefore the new price is $\\$\\var{sale_price}$.

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An item, originally priced at $\\$\\var{original_price}$, is on sale for $\\$\\var{sale_price}$.

The percentage discount must be [[0]]$\\%$.

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Note: Since the discount is off the original price, the discount percentage is a percentage of the original price.

You should understand both approaches below (even though approach 2 might be more obvious):

Approach 1

We can determine what proportion of the original price the new price is, by doing

$\\dfrac{\\text{new price}}{\\text{original price}}=\\dfrac{\\var{sale_price}}{\\var{original_price}}=\\var{remainder_decimal}=\\var{remainder_percent}\\%$

and we can say:

\"The sale price is $\\var{remainder_decimal}$ of the original\" or

\"The sale price is $\\var{remainder_percent}\\%$ of the original.\"

This means that the discount percentage must have been $100\\%-\\var{remainder_percent}\\%=\\var{discount_percent}\\%$.

Approach 2

The discount results is a saving of $\\$\\var{original_price}-\\$\\var{sale_price}=\\$\\var{price_difference}$.

Of the original price, this represents $\\dfrac{\\var{price_difference}}{\\text{original price}}=\\dfrac{\\var{price_difference}}{\\var{original_price}}=\\var{discount_decimal}=\\var{discount_percent}\\%$

and therefore the discount percentage was $\\var{discount_percent}\\%$.

An item, originally priced at $\\$\\var{original_price}$, has its price increased to $\\$\\var{sale_price}$.

The price was increased by [[0]]$\\%$.

Note: Since the increase is to the original price, the percentage increase is a percentage of the original price.

You should understand both approaches below (even though approach 2 might be more obvious):

Approach 1

We can determine what proportion of the original price the new price is, by doing

$\\dfrac{\\text{new price}}{\\text{original price}}=\\dfrac{\\var{sale_price}}{\\var{original_price}}=\\var{remainder_decimal}=\\var{remainder_percent}\\%$

and we can say:

\"The new price is $\\var{remainder_decimal}$ of the original\" or

\"The new price is $\\var{remainder_percent}\\%$ of the original.\"

This means that the percentage increase must have been $\\var{remainder_percent}\\%-100\\%=\\var{discount_percent}\\%$.

Approach 2

The increase was of $\\$\\var{sale_price}-\\$\\var{original_price}=\\$\\var{price_difference}$.

and therefore the percentage increase was $\\var{discount_percent}\\%$.

An item has a sale price of $\\$\\var{sale_price}$ during an $\\var{discount_percent}\\%$ off sale.

What was the item's original price? $\\$$ [[0]]

Note, you can't just add $\\var{discount_percent}\\%$ of $\\$\\var{sale_price}$ onto $\\$\\var{sale_price}$ since it was $\\var{discount_percent}\\%$ of the unknown original price that was taken away.

You should understand all three approaches below (and convince yourself that the final calculations in each method are all equivalent):

Unitary approach

We know that $\\var{remainder_percent}\\%$ of the original amount is $\\$\\var{sale_price}$.

This means that $1\\%$ of the original amount is $\\$\\dfrac{\\var{sale_price}}{\\var{remainder_percent}}$ (by dividing both amounts by $\\var{remainder_percent}$).

And that $100\\%$ of the original amount is $\\$\\dfrac{\\var{sale_price}}{\\var{remainder_percent}}\\times 100$ (by multiplying both amounts by $100$).

Working this out (with or without a calculator) we find the original price is $\\$\\var{original_price}$.

Scaling approach

We know that $\\var{remainder_percent}\\%$ of the original amount is $\\$\\var{sale_price}$.

To scale/stretch $\\var{remainder_percent}\\%$ into $100\\%$ we need to multiply it by $\\dfrac{100}{\\var{remainder_percent}}$.

This gives $100\\%$ of the original amount is $\\$\\var{sale_price}\\times \\dfrac{100}{\\var{remainder_percent}}$ (by multiplying both amounts by $\\dfrac{100}{\\var{remainder_percent}}$).

Working this out (with or without a calculator) we find the original price is $\\$\\var{original_price}$.

Algebraic approach

We know that $\\var{remainder_percent}\\%$ of the original amount is $\\$\\var{sale_price}$.

We could think of this as the equation $\\var{remainder_decimal}\\times \\text{Original}=\\var{sale_price}$.

Since division is the inverse of multiplication, we can divide both sides of this equation by $\\var{remainder_decimal}$ to get

$\\text{Original}=\\dfrac{\\var{sale_price}}{\\var{remainder_decimal}}$.

Working this out (with or without a calculator) we find the original price is $\\$\\var{original_price}$.

An item has a new price of $\\$\\var{sale_price}$ after an $\\var{discount_percent}\\%$ price increase.

What was the item's original price? $\\$$ [[0]]

Note, you can't just subtract $\\var{discount_percent}\\%$ of $\\$\\var{sale_price}$ off $\\$\\var{sale_price}$ since it was $\\var{discount_percent}\\%$ of the unknown original price that was added.

You should understand all three approaches below (and convince yourself that the final calculations in each method are all equivalent):

Unitary approach

We know that $\\var{remainder_percent}\\%$ of the original amount is $\\$\\var{sale_price}$.

This means that $1\\%$ of the original amount is $\\$\\dfrac{\\var{sale_price}}{\\var{remainder_percent}}$ (by dividing both amounts by $\\var{remainder_percent}$).

And that $100\\%$ of the original amount is $\\$\\dfrac{\\var{sale_price}}{\\var{remainder_percent}}\\times 100$ (by multiplying both amounts by $100$).

Working this out (with or without a calculator) we find the original price is $\\$\\var{original_price}$.

Scaling approach

We know that $\\var{remainder_percent}\\%$ of the original amount is $\\$\\var{sale_price}$.

To scale/shrink $\\var{remainder_percent}\\%$ into $100\\%$ we need to multiply it by $\\dfrac{100}{\\var{remainder_percent}}$.

This gives $100\\%$ of the original amount is $\\$\\var{sale_price}\\times \\dfrac{100}{\\var{remainder_percent}}$ (by multiplying both amounts by $\\dfrac{100}{\\var{remainder_percent}}$).

Working this out (with or without a calculator) we find the original price is $\\$\\var{original_price}$.

Algebraic approach

We know that $\\var{remainder_percent}\\%$ of the original amount is $\\$\\var{sale_price}$.

We could think of this as the equation $\\var{remainder_decimal}\\times \\text{Original}=\\var{sale_price}$.

Since division is the inverse of multiplication, we can divide both sides of this equation by $\\var{remainder_decimal}$ to get

$\\text{Original}=\\dfrac{\\var{sale_price}}{\\var{remainder_decimal}}$.

Working this out (with or without a calculator) we find the original price is $\\$\\var{original_price}$.

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A number is {first} by $\\var{percent}\\%$ and then {second} by $\\var{percent}\\%$, the result is a number that is [[0]] the original number.

In particular, the result is [[1]]$\\%$ of the original number.

In other words, a [[2]] of [[3]]$\\%$.

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Your initial instinct might be that increasing and decreasing by the same percentage results in no net change to the original number. That's an easy mistake to make. Here are two ways we can work out the truth of the matter:

Example approach

Start with 100 as the original number (as it is easy to work with for percentages).

Now {first_ing} by $\\var{percent}\\%$ leaves you with $\\var{example_1}$.

Then {second_ing} by $\\var{percent}\\%$ (of the current amount, $\\var{example_1}$), leaves you with $\\var{factor_percent}$.

Therefore, the two changes led to $\\var{factor_percent}\\%$ of the original, a decrease of $\\var{decrease_percent}\\%$.

Multiplication approach

Whatever the original number is, {first_ing} it by $\\var{percent}\\%$ means multiplying by $\\var{mult_1}$, and {second_ing} by $\\var{percent}\\%$ means multiplying by $\\var{mult_2}$.

But multiplying by $\\var{mult_1}$ and then by $\\var{mult_2}$ is equivalent to multiplying by $\\var{mult_1}\\times\\var{mult_2}=\\var{factor_decimal}$.

Therefore, the two changes led to $\\var{factor_percent}\\%$ of the original, a decrease of $\\var{decrease_percent}\\%$.

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