LAT Mathematics Syllabus

What does percentage mean and how do you convert a fraction to a percentage?

Percentage means 'per hundred' (out of 100). To convert a fraction or decimal to a percentage, multiply it by 100 and add the % sign. Example: 3/4 = 0.75 × 100 = 75%.

How do you find x% of a quantity Q?

x% of Q = (x/100) × Q. For example, 20% of 150 = (20/100) × 150 = 30.

How do you express one number as a percentage of another?

Percentage = (Part ÷ Whole) × 100. Example: 30 as a percentage of 120 is (30/120) × 100 = 25%.

What is the formula for percentage increase and percentage decrease?

Percentage change = (Change in value ÷ Original value) × 100. It is an increase if the new value is higher and a decrease if it is lower.

If a quantity is increased by x% and then decreased by x%, what is the net change?

There is a net decrease, not zero. The net percentage change is −(x²/100)%. For example, +10% then −10% gives a net change of −1%.

What is a ratio and how is it written?

A ratio is a comparison of two quantities of the same kind by division, showing how many times one is of the other. It is written as a:b (read 'a to b'), equivalent to the fraction a/b. Ratios should be expressed in simplest form.

What is a proportion and what is the cross-multiplication property?

A proportion states that two ratios are equal: a:b = c:d (or a/b = c/d). The cross-multiplication (product of means equals product of extremes) gives a × d = b × c.

How do you divide a quantity in the ratio a:b?

Add the ratio parts (a+b) to get total parts. Each part = quantity ÷ (a+b). First share = a × (quantity/(a+b)); second share = b × (quantity/(a+b)).

What is the difference between direct proportion and inverse proportion?

In direct proportion, two quantities increase or decrease together so their ratio stays constant (y = kx). In inverse proportion, one increases as the other decreases so their product stays constant (xy = k, i.e. y = k/x).

In a proportion a:b = c:d, what are the means and the extremes?

The extremes are the first and last terms (a and d). The means are the two middle terms (b and c). The product of the means equals the product of the extremes.

How are Cost Price (CP), Selling Price (SP), Profit and Loss defined?

CP is the price at which an item is bought; SP is the price at which it is sold. If SP > CP there is a Profit = SP − CP. If SP < CP there is a Loss = CP − SP.

What are the formulas for Profit% and Loss%?

Profit% = (Profit ÷ CP) × 100 and Loss% = (Loss ÷ CP) × 100. Both percentages are always calculated on the Cost Price.

How do you find SP given CP and a profit% or loss%?

On profit: SP = CP × (100 + Profit%)/100. On loss: SP = CP × (100 − Loss%)/100.

How do you find CP given SP and a profit% or loss%?

On profit: CP = SP × 100/(100 + Profit%). On loss: CP = SP × 100/(100 − Loss%).

What do 'marked price' and 'discount' mean, and how is discount calculated?

The marked price (MP, or list price) is the price labelled on an item before any reduction. A discount is a reduction given on the marked price: Discount = MP − SP, and Discount% = (Discount ÷ MP) × 100. Selling price = MP − Discount.

What is a simple (linear) equation and what does solving it mean?

A simple equation is an equation with one variable raised to the first power only (e.g. 2x + 3 = 11). Solving it means finding the value of the variable that makes both sides equal.

What is the golden rule for solving a simple equation?

Whatever operation you perform on one side of the equation, you must perform the same operation on the other side, so the equation stays balanced. Use inverse operations to isolate the variable.

How do you transpose a term across the equals sign in an equation?

When a term is moved to the other side of the equation, its sign/operation is reversed: + becomes −, − becomes +, × becomes ÷, and ÷ becomes ×. Example: x + 5 = 12 → x = 12 − 5.

How do you solve the equation ax + b = c for x?

Subtract b from both sides to get ax = c − b, then divide both sides by a to get x = (c − b)/a (provided a ≠ 0).

What is a term, a coefficient, a variable and a constant in an algebraic expression?

A variable is a symbol representing an unknown (e.g. x). A coefficient is the numerical factor multiplying a variable (in 5x, 5 is the coefficient). A constant is a fixed number with no variable. A term is a single number or variable, or numbers and variables multiplied together, separated by + or − signs.

What are like terms and unlike terms, and which can be combined?

Like terms have exactly the same variables raised to the same powers (e.g. 3x and 5x). Unlike terms have different variables or powers (e.g. 3x and 3x²). Only like terms can be added or subtracted (combined).

What are the algebraic identities for (a+b)², (a−b)² and (a+b)(a−b)?

(a+b)² = a² + 2ab + b²; (a−b)² = a² − 2ab + b²; and (a+b)(a−b) = a² − b² (difference of two squares).

What are monomials, binomials and trinomials?

They are polynomials classified by the number of terms: a monomial has one term (e.g. 5x), a binomial has two terms (e.g. x + 3), and a trinomial has three terms (e.g. x² + 2x + 1).

What is the basic relationship between speed, distance and time?

Speed = Distance ÷ Time. Rearranged: Distance = Speed × Time, and Time = Distance ÷ Speed.

How do you convert a speed from km/h to m/s and from m/s to km/h?

To convert km/h to m/s, multiply by 5/18. To convert m/s to km/h, multiply by 18/5.

How are relative speeds calculated for two objects moving toward each other versus in the same direction?

When two objects move toward each other (or in opposite directions), relative speed = sum of their speeds. When they move in the same direction, relative speed = difference of their speeds.

In a time-and-work problem, how do you express one person's rate of work?

If a person can complete a job in n days, their one-day work (rate) is 1/n of the job. The whole job is taken as 1 unit of work.

If A can do a job in x days and B in y days, how long do they take working together?

Their combined one-day work is 1/x + 1/y. Time taken together = 1 ÷ (1/x + 1/y) = xy/(x+y) days.

What is the formula for Simple Interest, and what do the variables stand for?

Simple Interest SI = (P × R × T) / 100, where P is the principal (amount invested/borrowed), R is the rate of interest per annum (%), and T is the time in years.

In simple interest, what is the Amount (total) and how does it relate to principal and interest?

Amount (A) = Principal + Simple Interest = P + (P × R × T)/100 = P(1 + RT/100). The amount is the total money to be repaid or received at the end of the period.