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2022 AMC 10A Problem 4

2022 AMC 10A Problem 4AMC 10 Step by Step

Name: Date: September 13, 2026

Ratios, Percents & Averages · difficulty 2 of 5 · about 3 min · position P1-10

In some countries, automobile fuel efficiency is measured in liters per $100$ kilometers while other countries use miles per gallon. Suppose that 1 kilometer equals $m$ miles, and $1$ gallon equals $l$ liters. Which of the following gives the fuel efficiency in liters per $100$ kilometers for a car that gets $x$ miles per gallon?

A)
$\frac{x}{100lm}$
B)
$\frac{xlm}{100}$
C)
$\frac{lm}{100x}$
D)
$\frac{100}{xlm}$
E)
$\frac{100lm}{x}$

Answer

Key insight

Convert 100 kilometers to 100m miles, divide by x miles per gallon to get gallons used, then multiply by l liters per gallon.

Solution

We want the number of liters the car burns while driving $100$ kilometers.

Distance: $100$ kilometers is $100m$ miles.

Fuel in gallons: the car covers $x$ miles per gallon, so $100m$ miles use $\dfrac{100m}{x}$ gallons.

Fuel in liters: each gallon is $l$ liters, so the trip uses $\dfrac{100m}{x} \cdot l = \dfrac{100lm}{x}$ liters.

The answer is $\boxed{\textbf{(E)}\ \frac{100lm}{x}}$ .

Why this works

Chain the conversions one unit at a time: kilometers to miles, miles to gallons (dividing by an efficiency), gallons to liters. Tracking units confirms the form: $\frac{\text{km} \cdot (\text{mi}/\text{km})}{\text{mi}/\text{gal}} \cdot \frac{\text{L}}{\text{gal}} = \text{L}$ . Notice that a "per gallon" figure must be divided by, which is why $x$ lands in the denominator.

Alternative approach

Plug in easy numbers: $m = 2$ , $l = 1$ , $x = 50$ . Then $100$ km is $200$ miles, which uses $4$ gallons, i.e. $4$ liters. Evaluate the choices: (A) $\frac{1}{4}$ , (B) $1$ , (C) $\frac{1}{2500}$ , (D) $1$ , (E) $4$ . Only (E) matches.

The trap

Inverting the rate: liters per 100 km measures consumption, the reciprocal of an efficiency like miles per gallon, so x must end up in the denominator.

Common mistakes

Inverting the rate: liters per 100 km measures consumption, the reciprocal of an efficiency like miles per gallon, so x must end up in the denominator.
Placing the factor $100$ in the denominator (choices (A), (B), (C)) by treating "per $100$ km" as division by $100$ rather than a distance of $100$ km.

Techniques

Set up the equation/formula and compute; no special trick needed · Test small/specific values or special cases to find or verify the answer