4.3: Real-World Problems Algebraic Equations

Solve a word problem using bar models Lisa wrote a riddle: A positive number is 1 3 of another positive number. If their difference is 48, find the two positive numbers. 2 units→ 48 1 unit → 48 2 = 24 3 units → 24 · 3 = 72 Second Number First Number The two positive numbers are 72 and 24.

Solve a word problem using algebraic reasoning Lisa wrote a riddle: A positive number is 1 3 of another positive number. If their difference is 48, find the two positive numbers. Let one number be represented by 𝑥. The other number is 1 3 of the first number so let the other number be represented by 1 3 𝑥. 𝑥− 1 3 𝑥=48 3 3 𝑥− 1 3 𝑥=48 2 3 𝑥=48 3 2 ∙ ∙ 3 2 𝑥=72 Second Number If one number is 72, the other number is 1 3 ∙72=24 . First Number The two positive numbers are 72 and 24.

Solve a word problem using bar models Mark wrote a riddle: A negative number is 2 5 of another negative number. If the sum of the two negative numbers is −35, find the two negative numbers. 7 units→ −35 1 unit → −35 7 = −5 2 units → −5 ∙2=−10 5 units → −5 ∙5=−25 First Number Second Number The two negative numbers are −10 and −25.

Solve a word problem using algebraic reasoning Mark wrote a riddle: A negative number is 2 5 of another negative number. If the sum of the two negative numbers is −35, find the two negative numbers. Let one number be represented by 𝑥. The other number is 2 5 of the first number so let the other number be represented by 2 5 𝑥. 𝑥+ 2 5 𝑥=−35 5 5 𝑥+ 2 5 𝑥=−35 7 5 𝑥=−35 5 7 ∙ ∙ 5 7 𝑥=−25 Second Number If one number is -25, the other number is 2 5 ∙ −25 =−10. First Number The two positive numbers are −25 and −10.

Solve a real-world problem using bar models and algebraic reasoning Jared has 12 more comic books than Zoe. If they have 28 comic books altogether, find the number of comic books Jared has. 𝑥+12 +𝑥= 28 2𝑥+12= 28 −12 −12 2𝑥= 16 Zoe has 8 comic books. Jared has 12 more than Zoe, so Jared has 8+12 = 20 comic books. 2 2 𝑥= 8

Solve a real-world problem using bar models and algebraic reasoning James has 16 more game cards than Fay. If they have 48 game cards altogether, find the number of game cars James has. 𝑥+16 +𝑥= 48 2𝑥+16= 48 −16 −16 2𝑥= 32 Fay has 16 game cards. James has 16 more than Fay, so James has 16+16 = 32 game cards. 2 2 𝑥= 16

Solve a real- world problem using a table A theater is divided into a red section and a blue section. The red section has 350 seats, and the rest of the seats are in the blue section. A ticket for a red section seat costs $75, and a ticket for a blue section seat costs $50. a) Write an expression for the total amount collected from the sale of tickets for all the seats in the two sections. Type of Ticket Red Blue Ticket Price ($) 75 50 # of Tickets 350 x Ticket Sales ($) 75·350= 26,250 50·x= 50x 26250+50𝑥

Solve a real- world problem using a table A theater is divided into a red section and a blue section. The red section has 350 seats, and the rest of the seats are in the blue section. A ticket for a red section seat costs $75, and a ticket for a blue section seat costs $50. b) The total sales when all the tickets are sold are $68,750. How many seats are in the blue section? Type of Ticket Red Blue Ticket Price ($) 75 50 # of Tickets 350 x Ticket Sales ($) 75·350= 26,250 50·x= 50x 26250+50𝑥= 68750 −26250 −26250 50𝑥=42500 50 50 𝑥=850 There are 850 seats in the blue section.