Examkin

Soru

Zorluk: OrtaRatio and Proportion Word Problems

A regional logistics company maintains a fleet consisting of delivery vans, medium trucks, and heavy cargo trucks in the initial ratio of 3:4:53 : 4 : 5, respectively. After the company retires 4 delivery vans, purchases 12 medium trucks, and purchases 6 heavy cargo trucks, the ratio of delivery vans to heavy cargo trucks becomes 1:21 : 2. What was the total number of vehicles in the fleet initially?

  1. A
    120
  2. B
    144
  3. 168Cevap
  4. D
    182
  5. E
    196

Cevap

168
The initial ratio of 3:4:53 : 4 : 5 implies that the number of vans, medium trucks, and heavy cargo trucks can be represented as 3x3x, 4x4x, and 5x5x for some positive multiplier xx. The initial total fleet is 3x+4x+5x=12x3x + 4x + 5x = 12x. Modifying the counts according to the problem gives 3x−43x - 4 vans and 5x+65x + 6 heavy trucks. Equating their ratio to 1/21/2 produces 3x−45x+6=12\frac{3x - 4}{5x + 6} = \frac{1}{2}. Cross-multiplying gives 6x−8=5x+66x - 8 = 5x + 6, which simplifies to x=14x = 14. Multiplying 1414 by the initial sum of ratio units (1212) yields the correct initial total of 168168.

Adım Adım Çözüm

1
Define initial quantities using a common multiplier xx
Delivery vans =3x= 3x, Medium trucks =4x= 4x, Heavy cargo trucks =5x= 5x. Initial total fleet =3x+4x+5x=12x= 3x + 4x + 5x = 12x.
Expressing each vehicle count in terms of xx preserves the given initial ratio of 3:4:53 : 4 : 5.
2
Formulate the equation based on the updated vehicle counts and new ratio
New number of vans =3x−4= 3x - 4. New number of heavy cargo trucks =5x+6= 5x + 6. Equation: 3x−45x+6=12\frac{3x - 4}{5x + 6} = \frac{1}{2}.
The problem states that after retiring 4 vans and adding 6 heavy cargo trucks, the ratio of vans to heavy cargo trucks becomes 1:21 : 2.
3
Solve the algebraic equation for xx
2(3x−4)=1(5x+6)  ⟹  6x−8=5x+6  ⟹  x=142(3x - 4) = 1(5x + 6) \implies 6x - 8 = 5x + 6 \implies x = 14.
Cross-multiplying eliminates the fraction and isolates xx.
4
Calculate the initial total number of vehicles
Initial total =12x=12×14=168= 12x = 12 \times 14 = 168.
Substituting x=14x = 14 into the initial total expression 12x12x yields the total fleet size prior to any modifications.

Anahtar Kavram

Multi-part ratio scaling and structural algebraic modeling under quantity changes
Bu soruyu puanla

Konular

Ratio and Proportion Word Problems

Bu konudan daha fazla soru