8.1.21 Working Backwards

In some situations, you may be given the total after the discount or tax has been applied and asked to find the original amount. For example, suppose an online store charges an extra 5% of the cost of the purchases for postage and packing. You have a gift voucher for $30. What is the maximum you can spend at the store, so that the total including the postage is less than $30?

As you can see in the diagram above, 105% is equivalent to $30.

Therefore, 1% is equivalent to equation left hand side dollar times 30 division 105 almost equals right hand side dollar times 0.2857, and multiplying this quotient by 100 creates 100%, which results in equation left hand side 100 multiplication dollar 30 divided by 105 almost equals right hand side dollar 28.57 (rounded down to the nearest cent).

The maximum amount that can be spent then is $28.57. You can check your answer by working out the 5% postage charge on $28.57 to see that the grand total will be $30.

Activity symbolActivity: Working Backwards

The price of a tennis racquet, including a 4% sales tax, is $45.24. What was the price before sales tax?

Hint symbol


The $45.24 includes sales tax. What percentage does this total represent? Find the dollar amount that represents 1%, and use that figure to arrive at 100%.

Solution symbol


104% of the cost is $45.24.

1% is dollar 45.24 division 104 and 100% is equation left hand side 100 multiplication dollar 45.24 divided by 104 equals right hand side dollar 43.50.

Thus, the price of the tennis racquet before sales tax was applied was $43.50.

8.1.20 Going Down, Going Up