0580 · 2.8Proportion

Direct proportion and constants

0:00Buying more identical notebooks increases the total cost, provided the price per notebook stays fixed. Sharing a fixed task among more equally productive workers reduces the time needed. Both situations involve proportion, but their relationships work differently. By the end, you will understand how to express these relationships algebraically and use them to find unknown quantities. You need substitution, simple equation solving, and the meanings of squares, cubes and roots.

0:26Squaring means raising a number to the power of two, while a square root reverses that operation. Cubing uses the power of three, and a cube root reverses cubing. We will use positive quantities in our examples, so the roots have straightforward positive values. For the notebooks, doubling the number bought doubles the cost. Multiplying the number by any factor multiplies the cost by that same factor. We say the cost is directly proportional to the number of notebooks.

0:52The essential feature is a constant ratio, not simply that both quantities increase. Call the number of notebooks x and the cost y. Dividing y by x always gives three, because each notebook costs three pounds. In general, direct proportion means y ÷ x equals a constant called k. Multiplying both sides by x gives y = k × x. The symbol on screen is read as is proportional to.

1:15Writing y is proportional to x describes the relationship, but does not specify the constant. To calculate particular values, replace that symbol with an equals sign and introduce k. This number is the constant of proportionality, and its value stays unchanged within the given relationship. Here is a first exam-style example. The quantity y is directly proportional to x, and y = 18 when x = 6. We begin by writing y = k × x.

1:42Substituting the known values gives 18 = 6 × k. Dividing eighteen by six gives k = 3. Our equation is therefore y = 3 × x. To find y when x = 11, substitute eleven into that equation. This gives y = 3 × 11. The answer is thirty-three, which is sensible because the same constant multiplier connects both pairs of values.

Inverse proportion and constant products

2:06For the fixed task, assume every worker works at the same rate without getting in anyone else’s way. Doubling the workers then halves the time required. Tripling their number divides the time by three. This is inverse proportion, where multiplying one quantity by a factor divides the other by that factor. The number of workers multiplied by the time remains constant. With two workers taking six hours, that product is twelve.

2:30Algebraically, inverse proportion means y = k ÷ x, so x × y = k. We also write y is proportional to 1 ÷ x. Inverse proportion does not mean a negative constant or a negative answer. The word inverse describes the reciprocal relationship, with x in the denominator. Also, a quantity decreasing as another increases is not enough to establish inverse proportion. The product must remain constant for the linear inverse relationship we have just described.

Powers, roots and proportion

2:58Proportion can involve a power or root of a quantity, rather than the quantity itself. A square’s area is directly proportional to its side length squared. Doubling the side length multiplies the area by four, because both dimensions double. The constant ratio is now area divided by side length squared, not area divided by side length. If y is directly proportional to x², write y = k × x². If x triples, its square becomes nine times as large, so y is also multiplied by nine.

3:28For direct cube proportion, doubling x multiplies y by eight. A cube has three dimensions, so doubling each dimension explains that factor of eight. The same idea applies to square roots and cube roots. If y is directly proportional to √x, multiplying x by four doubles y. For direct cube root proportion, multiplying x by eight doubles y. These changes follow from the roots of four and eight, rather than from the original multipliers.

3:54For each inverse version, put the stated power or root in the denominator. So inverse square proportion gives y = k ÷ x², and the square root version puts √x underneath instead. The matching cube and cube root relationships use those expressions in the denominator instead. This gives a useful way to check answers before calculating. Doubling x divides y by four in inverse square proportion, or by eight in inverse cube proportion.

4:20Multiplying x by four halves y in inverse square root proportion. For inverse cube root proportion, multiplying x by eight halves y. Suppose y is inversely proportional to x², and y = 20 when x = 3. Write y = k ÷ x². Substitution gives 20 = k ÷ 3². Since 3² is nine, this becomes 20 = k ÷ 9. Multiplying twenty by nine gives k = 180. Our equation is y = 180 ÷ x².

4:55When x = 6, the denominator is 6², which equals thirty-six. Dividing one hundred and eighty by thirty-six gives y = 5. Doubling x has divided y by four, exactly as expected. Our next example asks for the quantity inside a root. The quantity y is directly proportional to √x. We know y = 15 when x = 25. Start with y = k × √x. Substituting gives 15 = k × √25.

5:25√25 is five, so 15 = 5 × k. Dividing fifteen by five gives k = 3. The equation becomes y = 3 × √x. When y = 24, we therefore have 24 = 3 × √x. Divide twenty-four by three to obtain √x = 8. Squaring both sides gives x = 64. We square only after isolating the root, because that makes the reversal clear. Checking gives 3 × √64, which equals twenty-four.

Inverse cube roots and exam method

5:59For a final exam-style example, y is inversely proportional to the cube root of x. We know y = 6 when x = 8. Write y = k divided by the cube root of x. Substituting gives 6 = k divided by the cube root of eight. That root is two, so 6 = k ÷ 2. 6 × 2 gives k = 12. The equation is now y = 12 divided by the cube root of x.

6:25When y = 3, we have 3 = 12 divided by the cube root of x. Multiplying both sides by that root gives three times the cube root of x = 12. Dividing twelve by three gives the cube root of x = 4. Cubing both sides produces x = 64. Here, y has halved while x has increased from eight to sixty-four. That eightfold increase matches inverse cube root proportion and confirms that the answer fits the relationship.

6:52A common mistake is treating every direct relationship as y = k × x. Read exactly what y is proportional to, including any square, cube or root. Another mistake is using y = k × x² for inverse square proportion. The square must be in the denominator because increasing it must reduce y. Increasing together does not automatically mean directly proportional. For example, y = x + 3 increases with x, but y ÷ x is not constant.

7:21Likewise, decreasing together with increasing x does not prove inverse proportion. Check the appropriate ratio or product, rather than relying only on the direction of change. An examination may ask you to express a relationship algebraically, find the constant, or calculate either unknown quantity. Show the model first, then substitute the known pair to find k. Use the completed equation for the requested value, and show any rearrangement. These steps make your reasoning clear instead of leaving an unexplained answer.

7:50These methods must work without a calculator on Paper two, including familiar powers and roots. Paper four allows a calculator, but the algebraic model stays the same. Keep exact values throughout the working and avoid rounding intermediate results. Unless instructed otherwise, give answers that are not exact to three significant figures. The central idea is that proportion preserves a constant relationship. Direct proportion keeps a ratio constant, while inverse proportion keeps the corresponding product constant.

8:18Identify the exact power or root, write an equation with k, and find that constant first. Then calculate the unknown and check that its change agrees with the relationship.