How many km does 1cm on a map actually represent, and is this sugar water too sweet or too weak? Two completely different-looking situations, but they're explained by the exact same math idea: a ratio. A ratio compares two quantities by division (e.g. sugar:water = 1:9), and a "proportion" (as a rate) expresses that ratio as a single value — a fraction or a percentage.
Ratios have one very important property: multiplying or dividing both numbers by the same number never changes the ratio itself. Mix 20g of sugar with 180g of water for a 10% concentration sugar water — double both sides to 40g of sugar and 360g of water, and the concentration is still 10%. This property is exactly what lets you scale an amount up or down while keeping the same taste. It's also why a recipe says "double every ingredient to go from 2 servings to 4."
A map's scale works the same way. A scale of 1:25,000 means "1cm on the map represents 25,000cm (=250m) in reality." Because this ratio stays constant across the whole map, measuring the distance between two points on the map with a ruler is all you need to calculate the real-world distance. The bigger the scale's second number (e.g. 1:100,000), the wider an area a single map can cover, though fine details get blurry; the smaller it is (e.g. 1:5,000), the more closely a small area is shown in detail. That's exactly why hiking maps and detailed maps use a small scale number, while a map of an entire country uses a large one.
Ratio properties show up all over daily life. When a photo lab enlarges a photo, if the width-to-height ratio doesn't match the original, a person's face ends up looking stretched long or squashed flat — because the ratio got broken. Architectural blueprints and model cars also have to keep exactly the same ratio as the real thing so the actual size can be calculated, and the ratio of sugar to salt in a recipe or the ratio of colors when mixing paint all follow this same idea. Even currency exchange calculations are really just using the ratio between two currencies — understanding this one principle applies to a surprisingly wide range of fields.
When studying this with kids, it helps a lot to have an activity that lets them see with their own eyes that "the relationship stays the same even when you multiply or divide both numbers by the same number." Make two cups of sugar water, double the ingredients for one of them, and have them actually taste both (and notice the sweetness feels the same) — that sticks in memory far longer than learning it as numbers alone. For map scale, it's a great activity to actually measure the distance from home to school on a map app, then compare it with the value calculated by hand using the scale.
On our activity page, you can switch between two situations with tabs: map scale and sugar water concentration. In the sugar water tab, click "double it" and see for yourself that the concentration stays exactly the same even as the amount grows. In the map tab, change the scale and compare how much the real-world distance changes even for the same distance on the map.