Teach: Algebraic Expressions ② Representing Quantities with Letters

Algebraic Expressions ② Representing Quantities with Letters

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）.

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）
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

  1. Express the total cost of a snacks costing 90 yen each and b juices costing 120 yen each using an algebraic expression.
  2. Express the time needed to travel 40 km by a car traveling at x km per hour using an algebraic expression.
  3. 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

  1. 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.
  2. Answer：40/x（hours）
    Explanation：Time＝distance÷speed ＝ 40÷x ＝ 40/x。The unit is hours.
  3. 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.

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