Ratios and proportions

Express both lengths in the same unit. Simplify the plan-to-real length ratio so its first term is 1.

Total price = price of one item × number of items. Every item has the same unit price, so include that price once for each item.

Multiply the map distance in centimetres by the kilometres represented by one centimetre. For example, if 1 cm represents 4 km, then 3 cm represents 3 × 4 = 12 km.

Convert the actual distance to centimetres and divide by the scale denominator.

First divide the ingredient amount by the original number of servings to find the amount per serving. Multiply by the required servings: new amount = original amount ÷ original servings × new servings.

Write the ratio in the requested order: number of red balls : number of blue balls. Simplify both terms by a common divisor if needed. Do not compare the red count with the total number of balls.

For directly proportional quantities, y/x is constant. Find the unit value y ÷ x from a known pair, then multiply it by the x value of the incomplete pair. Check that all pairs have the same ratio.

Divide both ratio terms by their greatest common divisor. Keep their order. For example, 12:18 = 2:3 because both terms were divided by 6.

In a/b = x/d, the cross products are equal: a × d = b × x. Therefore x = a × d ÷ b. Alternatively, find the factor from b to d and multiply a by the same factor.

Divide the original total price by the original item count to find the unit price, then multiply by the new count. For example, if 3 items cost €12, then 5 cost 12 ÷ 3 × 5 = €20.