Letters can be used to express various quantities, such as prices, speeds, and areas, in expressions. It is helpful to think about “what happens when letters are substituted into a formula.” During the first year of middle school, become familiar with basic ways of expressing quantities.
Prices, Numbers of Items, and Ratios
- Cost of b apples at a yen each → ab yen
- Total cost of five notebooks at x yen each and three pens at y yen each → 5x + 3y yen
- Price of an item costing a yen after a 10% discount → 0.9a yen （subtract 10%＝1/10）
- x % of a people → a × x/100 = ax/100 people
Tip for ratios：Remember that “○%” means “○/100” and “○割” means “○/10.” x% of a is ax/100.
Example ①
Express the following quantities using algebraic expressions.
(1) Change received when buying five snacks costing x yen each and paying with a 1000-yen bill
(2) Selling price when an item with a list price of a yen is sold at a 20% discount
Answer：
(1) Cost 5x yen, change = 1000 − 5x → 1000 − 5x（yen）
(2) A 20% discount means 80% of the list price（0.8 times）→ 0.8a（yen）. Alternatively, 4a/5（yen）.
Express the following quantities using algebraic expressions.
(1) Change received when buying five snacks costing x yen each and paying with a 1000-yen bill
(2) Selling price when an item with a list price of a yen is sold at a 20% discount
Answer：
(1) Cost 5x yen, change = 1000 − 5x → 1000 − 5x（yen）
(2) A 20% discount means 80% of the list price（0.8 times）→ 0.8a（yen）. Alternatively, 4a/5（yen）.
Speed, Distance, and Time
Remember the three formulas（using the “distance–speed–time” image）.
Distance ＝ Speed × Time
Speed ＝ Distance ÷ Time
Time ＝ Distance ÷ Speed
- Distance traveled at a km per hour for t hours → at km
- Time needed to walk an x-km road at 4 km per hour → x/4 hours
- Distance traveled at 60 m per minute for y minutes → 60y m
Lengths, Areas, and Volumes of Shapes
- Square with side length a cm：perimeter 4a cm, area a² cm²
- Rectangle with length a cm and width b cm：perimeter 2(a+b) cm, area ab cm²
- Triangle with base a cm and height h cm：area ah/2 cm²
- Cube with side length a cm：volume a³ cm³
Example ②
A car traveling at x km per hour travels for 5 hours from Point A, then travels for another 2 hours at y km per hour.
Express the total distance using an algebraic expression.
Answer：First part ＝ 5x km, second part ＝ 2y km. Total ＝ 5x + 2y（km）
A car traveling at x km per hour travels for 5 hours from Point A, then travels for another 2 hours at y km per hour.
Express the total distance using an algebraic expression.
Answer：First part ＝ 5x km, second part ＝ 2y km. Total ＝ 5x + 2y（km）
Key Points
- Convert percentages to /100 and percentages expressed in tenths to /10 to make expressions easier to form
- Distance＝speed×time. Once one of the three formulas is learned, the others can be made by rearranging it
- Do not forget ÷2 in the area of a triangle（ah/2）
- “After a discount” can be found by either “total−discount amount” or “（1−discount rate）×list price”
Practice Problems
- Express the total cost of a snacks costing 90 yen each and b juices costing 120 yen each using an algebraic expression.
- Express the time needed to travel 40 km by a car traveling at x km per hour using an algebraic expression.
- Express the area of a triangle with base a cm and height b cm using an algebraic expression. Also find its area when a=6 and b=5.
Answers and Explanations
- Answer：90a + 120b（yen）
Explanation：Calculate unit price×number of items for each product and add them. The snacks cost 90a yen in total, and the juices cost 120b yen in total. - Answer：40/x（hours）
Explanation：Time＝distance÷speed ＝ 40÷x ＝ 40/x。The unit is hours. - Answer：Area ab/2 cm²。When a=6 and b=5, 6×5÷2 ＝ 15 cm²
Explanation：Area of a triangle＝（base×height）÷2 ＝ ab/2。Substitute the values of a and b and calculate specifically.