Variation:

Variation simply means how one quantity changes when another quantity changes. In direct variation, when one thing increases, the other increases too at the same rate. For example, if you buy rice at ₦500 per kg, the total cost varies directly with the number of kilograms you buy.

Inverse variation works the opposite way. When one quantity increases, the other decreases. If workers share a fixed amount of work, more workers means less work per person.

Joint variation combines both ideas—one quantity depends on two or more other quantities at once. For instance, the area of a rectangle varies jointly with its length and width.

The key is recognizing which type you're dealing with, then using the correct formula: direct is y = kx, inverse is y = k/x, and joint is y = kxz.

When one quantity depends on another, we call it variation. In direct variation, two quantities increase or decrease together at a constant rate. For example, if a tailor charges ₦5,000 per fabric length, the total cost varies directly with the number of lengths you buy. If you buy 2 lengths, you pay ₦10,000; buy 3 lengths and pay ₦15,000.

Partial variation is different because there's a fixed cost plus a variable cost. Many Nigerian businesses work this way. A transport company might charge ₦500 per kilometer plus a fixed ₦1,000 booking fee. So your fare depends partly on distance and partly on that fixed amount.

The mathematical expressions are simple: direct variation is y = kx, while partial variation is y = kx + c, where k is the constant and c is the fixed amount.

When something gets more expensive or cheaper, we calculate how much it changed using percentages. Percentage increase means the original amount grew, while percentage decrease means it shrunk.

To find percentage increase, subtract the original from the new amount, divide by the original, then multiply by 100. For example, if a loaf of bread costs ₦250 and increases to ₦300, the increase is ₦50. Dividing ₦50 by ₦250 gives 0.2, then multiplying by 100 gives 20% increase.

Percentage decrease works the same way. If your school fees drop from ₦50,000 to ₦40,000, that's a ₦10,000 decrease. Dividing ₦10,000 by ₦50,000 and multiplying by 100 gives 20% decrease.

Always use the original amount as your denominator, never the new amount. This is the most common mistake students make.

When one quantity decreases while another increases, we call this inverse variation or decrease in variation. The mathematical relationship means as one variable goes up, the other comes down in a proportional way. Think of it like this: when the price of tomatoes increases in Balogun Market, fewer people buy them, so the quantity sold decreases. The relationship between price and quantity follows the formula y = k/x, where k is a constant. If tomatoes cost ₦50 per basket, you might buy 10 baskets, but at ₦100 per basket, you'd buy only 5 baskets. This inverse relationship appears in real life constantly. Speed and time show this pattern too—drive faster and your journey time reduces. When solving these problems, identify your constant k first by substituting known values, then use it to find unknowns.

When we say one quantity varies with another, we're describing their relationship. In linear variation, if y varies directly with x, then y equals a constant times x, written as y = kx. Think of buying airtime: the cost varies directly with the number of minutes you purchase.

Quadratic variation works differently. When y varies as the square of x, we write y = kx². For example, the force needed to stop a moving vehicle varies as the square of its speed—doubling your car's speed requires four times the braking force to stop safely.

To solve these problems, find the constant k using given values, then use it to answer questions about other values. Always substitute your known pairs first.

When two quantities vary together, they follow specific mathematical relationships. Direct variation means when one increases, the other increases proportionally—like the cost of buying yams at the market. If one yam costs ₦200, then five yams cost ₦1,000. Inverse variation works oppositely: when one quantity increases, the other decreases. For example, if your transport fare depends on how many friends share costs, the more people sharing, the less each person pays.

Inequalities describe situations where quantities aren't equal. Instead of writing x = 5, you might write x > 5 or x < 5, meaning x is greater than or less than 5. In variation problems, inequalities help us understand limits and ranges. A shopkeeper might need to sell at least 50 items daily to make profit, written as x ≥ 50.

When you see an inequality like y > 2x + 1 or y ≤ x - 3, the graph shows you all the points that make that inequality true. Think of it like marking areas on a map where certain conditions are met. If the inequality uses > or <, you draw a dashed line because points on the line itself don't satisfy it. If it uses ≥ or ≤, you draw a solid line because those points do count. The shaded region represents your solution set—all the coordinate pairs that work.

Imagine a Lagos trader selling rice and beans. If she needs to make at least ₦5,000 profit daily, the region above the line representing ₦5,000 shows all valid combinations of quantities she can sell. Points below that line don't meet her target.