Quantitative Reasoning notes

MDCAT Analytical and Logical Reasoning

Quantitative reasoning uses numbers, algebra, percentages, ratios, geometry, statistics, graphs and inequalities to solve practical problems. The main skill is to translate words, tables and diagrams into mathematical statements and then check whether the answer is reasonable.

Number Properties and Prime Numbers

A prime number is a whole number greater than 1 that has exactly two positive factors: 1 and itself. The number 2 is the only even prime number. Every other prime number is odd.

When adding several odd numbers, the result is odd if the number of odd terms is odd, and even if the number of odd terms is even. This helps determine whether a set of prime numbers must contain 2.

Prime numbers include 2, 3, 5, 7, 11 and 13.
The only even prime number is 2.
The sum of five odd prime numbers is odd.
The sum of five prime numbers can be even only if one of them is 2, because the remaining four primes are odd.
In a set of five prime numbers with an even sum, the smallest number is 2.
A composite number has more than two positive factors.
The factors of a number divide it exactly without a remainder.

Fractions, Ratios and Proportions

Fractions can be compared by using a common denominator, converting them to decimals, or cross-multiplying. For positive fractions a/b and c/d, compare ad with bc.

A proportion states that two ratios are equal. Direct proportion means that both quantities change by the same multiplier. If 20 machines produce 1,240 printers, the number of printers per machine is constant.

To compare a/b and c/d, cross-multiply and compare ad with bc.
The larger fraction is not always the one with the larger numerator or denominator.
In direct proportion, y = kx, where k is the constant of proportionality.
If 20 machines produce 1,240 printers, one machine produces 62 printers at the same rate.
To produce 1,984 printers, 1,984 ÷ 62 = 32 machines are needed.
The number of additional machines is 32 − 20 = 12.
For example, 5/6 is larger than any listed fraction that is less than 5/6.

Percentages, Profit and Discount

A percentage means a fraction out of 100. Profit and loss questions require the cost price and selling price to be compared. A discount reduces the original price, while profit compares the selling price with the cost price.

Always calculate the total cost before finding a percentage. In a bus fare problem, compare the weekly cost without the card with the card price plus the discounted fares.

Percentage = part ÷ whole × 100.
Profit = selling price − cost price.
Loss = cost price − selling price.
Profit percentage = profit ÷ cost price × 100.
For 100 oranges bought for Rs. 350, selling at Rs. 48 per dozen gives a total selling price of 100 × 48 ÷ 12 = Rs. 400.
The profit is Rs. 50, so profit percentage = 50 ÷ 350 × 100 = 14.27% approximately.
Five bus tickets at $2.50 each cost $12.50. A $10 card leaves $2.50 for fares, so each fare is 20% of the original price, giving an 80% discount.
Card cost plus discounted fares must equal the normal weekly cost when the two choices cost the same.

Algebraic Equations and Word Problems

An equation states that two expressions have equal values. Solve it by performing the same operation on both sides. Combining like terms and defining variables clearly prevents mistakes.

For word problems, assign a variable to the unknown quantity. Write each payment or condition in terms of that variable, then use the remaining amount or total to form an equation.

From 4x + 6 = 18, subtracting 6 from both sides gives 4x = 12.
An equation with the same solution can be obtained by applying an equivalent operation to both sides.
If a player answers 40 questions, incorrect answers = 40 − correct answers.
For score = twice correct answers − incorrect answers, let correct answers be c: 2c − (40 − c) = 50.
This gives 3c − 40 = 50, so c = 30.
If salary is S, groceries costing one tenth leave 9S/10.
Car payment of 3/18 of the remaining amount equals one sixth of 9S/10, or 3S/20.
If rent is twice the car payment, then the amount left is S − S/10 − 3S/20 − 3S/10 = 9S/20. Setting this equal to $1,620 gives S = $3,600.

Functions, Linear Relations and Graphs

A function assigns exactly one output to each input. A table is linear when equal changes in x produce equal changes in y. Its equation can be written as y = mx + b, where m is the slope and b is the y-intercept.

A graph can provide the intercepts directly. The y-intercept is the point where the graph crosses the y-axis, so its x-coordinate is always 0.

A linear function has the form y = mx + b.
The slope m is change in y divided by change in x.
The y-intercept is (0, b).
For a graph crossing the y-axis at −4, the y-intercept is (0, −4).
A table showing a constant first difference in output represents a linear function.
A nonlinear function does not have a constant rate of change.
A fixed fee plus a daily charge is represented by c = daily charge × number of days + fixed fee.
For a tent costing $11 per day and a $10 insurance fee, the equation is c = 11d + 10.

Quadratic Equations, Absolute Value and Intercepts

A quadratic equation has the general form ax2 + bx + c = 0, where a is not zero. Its solutions are also called roots. If the roots are known, factors can be formed from expressions that become zero at those roots.

An absolute value represents distance from zero, so it is never negative. An equation involving absolute value usually produces two cases.

If x = 3/4 is a root, one factor is 4x − 3.
If x = −2/5 is a root, one factor is 5x + 2.
Therefore, factors that could occur are (4x − 3) and (5x + 2).
For |2x + 1| = 5, solve 2x + 1 = 5 or 2x + 1 = −5.
The two solutions are x = 2 and x = −3.
The distance between these solutions is |2 − (−3)| = 5.
The x-intercepts of a graph are the x-values for which y = 0.
For x2 + 5x + 4 = 0, factorization gives (x + 1)(x + 4) = 0. The x-intercepts are −1 and −4, and their distance is 3.

Geometry, Measurement and Density

Geometry questions require the correct formula and consistent units. For a cylinder, volume equals base area multiplied by height. For a cube, volume is the cube of its edge length.

Density relates mass and volume. If density and volume are known, mass is found by multiplying them.

Volume of a cylinder = area of base × height.
If cylinder volume is 432 cm3 and base area is 24 cm2, height = 432 ÷ 24 = 18 cm.
Volume of a cube = side3.
For a cube with edge 0.90 m, volume = 0.90 × 0.90 × 0.90 = 0.729 m3.
Density = mass ÷ volume.
Mass = density × volume.
For density 807 kg/m3 and volume 0.729 m3, mass = 807 × 0.729 = 588.303 kg, which rounds to 588 kg.
Use cubic metres with density in kg/m3, and cubic centimetres with measurements given in cm.

Inequalities and Feasible Values

An inequality compares quantities using <, >, ≤ or ≥. Unlike an equation, an inequality usually represents a range of possible values. When multiplying or dividing by a negative number, reverse the inequality sign.

For a ratio involving a positive numerator and a negative denominator, the result is negative. To find the greatest value, choose the combination that makes the negative result closest to zero.

If 3 ≤ x ≤ 5 and −4 ≤ y ≤ −1, then x is positive and y is negative.
For x/y, the greatest value is obtained using x = 3 and y = −4.
Thus, the greatest value is 3/(−4) = −3/4.
A staffing problem with x junior directors and y senior directors can be modelled by 640x + 880y ≤ 9,700.
The requirement of at least 10 staff members is x + y ≥ 10.
If junior directors must be at least three times senior directors, then x ≥ 3y.
If at least one senior director is required, then y ≥ 1.
Staff numbers are normally non-negative integers, because a fraction of a staff member is not possible.

Sequences, Analogies and Calendar Reasoning

Number analogies require finding the same operation or pattern from the first pair and applying it to the second pair. Several rules may be mathematically possible, so select the simplest rule consistent with the given relationship and answer choices.

Calendar problems use the fact that weekdays repeat every seven days. Count the starting day as day one when the question asks for the weekday on a particular date.

For 7 : 57 and 10 : ?, the rule n(n + 1) + 1 gives 7 × 8 + 1 = 57 and 10 × 11 + 1 = 111.
For 6 : 10 and 9 : ?, the rule is adding 4, so the answer is 13.
A week has 7 days, so weekdays repeat after every seven days.
A 30-day month has 29 days after its first day, and 29 leaves a remainder of 1 when divided by 7.
If a 30-day month begins on Thursday, the 30th falls one weekday after Thursday, which is Friday.
Always distinguish a date number from the number of elapsed days after the first date.

Statistics and Data Interpretation

Statistics summarises data using frequency, mean, median, mode, range, histograms and box-and-whisker plots. The mean uses every value, while the median depends on the middle position after the data are arranged in order.

A histogram displays frequencies for intervals. A box plot displays the minimum, first quartile, median, third quartile and maximum. Compare the same feature in both data sets before selecting a statement.

Frequency is the number of observations in a class or interval.
A histogram interval such as 10 to 20 usually includes values greater than or equal to 10 and less than 20.
For 23 ordered values, the median is the 12th value.
Mean = sum of values ÷ number of values.
A box plot median is the line inside the box.
If the median lines of two box plots have the same position, the medians are equal.
For seven values, the median is the fourth value after arranging them in ascending order.
For 43, 83, 54, 35, 77, x and y, values x = 41 and y = 59 give the ordered list 35, 41, 43, 54, 59, 77, 83.
The sum of that list is 392, so the mean is 392 ÷ 7 = 56 and the median is 54.
In a random sample of 200 cars, 3 defects give a rate of 3/200. Applied to 10,000 cars, the expected number is 10,000 × 3/200 = 150.

Key terms

Prime number: A whole number greater than 1 with exactly two positive factors, 1 and itself.
Composite number: A whole number greater than 1 with more than two positive factors.
Ratio: A comparison of two quantities by division.
Proportion: An equation stating that two ratios are equal.
Percentage: A quantity expressed as a fraction out of 100.
Profit: The amount by which selling price exceeds cost price.
Discount: A reduction from the original price.
Linear function: A function with constant rate of change, usually written y = mx + b.
Slope: The change in y divided by the change in x.
Y-intercept: The point where a graph crosses the y-axis.
Root: A value of the variable that makes an equation true, especially a polynomial equation.
Absolute value: The non-negative distance of a number from zero.
Density: Mass per unit volume.
Inequality: A mathematical statement comparing quantities using symbols such as ≤ or >.
Mean: The sum of all data values divided by the number of values.
Median: The middle value of ordered data, or the average of the two middle values when the number of values is even.
Frequency: The number of times a value or class interval occurs.
Box-and-whisker plot: A diagram showing the minimum, quartiles, median and maximum of a data set.

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Analytical and Logical Reasoning shortcuts

Choosing the stronger supporting argument

Choose the option that directly explains why the claim is true. Prefer relevant evidence with a clear link to the claim, not a statement that is merely related.

Identify the claim and ask, “Why would this make the claim true?”
Reject personal opinions, unrelated facts and extreme statements.
Example: “Teachers should be paid more” is supported by “They shape future generations.”

This shortcut does not apply when the option gives direct statistical or experimental evidence that is stronger than a general reason.

Finding the pattern in letter and number series

Check letters and numbers separately. Look for equal increases, alternating changes, multiplication, or a repeating pattern.

Convert letters to positions if needed, such as A = 1 and D = 4.
Find the change in the letters and in the numbers independently.
Example: A2, D4, G6, J8 has letters increasing by 3 and numbers by 2, so the answer is M10.

Do not assume a simple addition pattern if alternating or multiplication patterns fit all the given terms.

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