Reasoning · Grade 6-1 Ratio and Rate (1)

Problem

Concentration of a salt solution

Concentration is the salt as a percent of the whole salt water, salt included. (1) 25 g of salt in 125 g of salt water is what percent? (2) 200 g at 15 % holds how much salt? (3) 5 g of salt at 10 % sits in how much salt water?. Answer all three.
Your answer
How to solve
Strategy Analyze the Units — Three quantities are in play — grams of salt, grams of salt water, and a percent — and only two of them are amounts. Watching the units keeps them straight and stops the commonest mistake, dividing by the water instead of by the whole salt water. The quantity worth holding on to is the mass of salt: it is a real pile of solid that sits in the bowl and does not change when water is added or boiled away, while the concentration is only a ratio and shifts the moment the water does. So in every part I aim at the grams of salt first and treat the percent as the instruction telling me what fraction of the whole those grams make up. I draw each mixture as a bar cut into 100 equal hundredths so the percent can be seen, and for part (3), where the whole is the missing piece, I run the same relation backwards.
1STEP 1

Sort the three quantities and their units

The three quantities are salt, salt water and concentration.

(concentration) % = (salt in g)/(salt water in g) × 100
2STEP 2

Notice which quantity stays put

Write the relation in all three directions.

(salt in g) = (salt water in g) × (concentration)/100
3STEP 3

Part (1): divide the salt by the whole salt water

(1) divides salt by the whole: 20 %.

25/125 × 100 = 1/5 × 100 = 20 %
4STEP 4

Part (2): take the percent of the whole to get the salt

(2) takes 15 % of the whole: 30 g.

200 × 15/100 = 30 g
5STEP 5

Part (3): run it backwards to recover the whole

(3) runs it backwards: the whole is 50 g.

(salt water) × 10/100 = 5 → (salt water) = 5/10 × 100 = 50 g
6STEP 6

Check all three against the definition

Put back into the definition, all three check out.

30/200 × 100 = 15 %, 5/50 × 100 = 10 %
Answer
20 %, 30 g, 50 g
25 ÷ 125 = 0.2
Units land where they should: (1) is a percent, while (2) and (3) are masses in grams, matching what each question asks for. Every concentration sits between 0 % and 100 %, and in each mixture the salt is lighter than the salt water that contains it — 25 g inside 125 g, 30 g inside 200 g, 5 g inside 50 g. The magnitudes also move the right way: part (3) asks for a whole and returns 50 g, ten times the 5 g of salt, which is exactly what a 10 % concentration should mean, and a wrong division there would have produced a total smaller than the salt itself.
Takeaway

The salt is the part that stays put — track its grams, and the percent tells you everything else about the mixture.

  • Sort the three quantities and their units
  • Notice which quantity stays put
  • Part (1): divide the salt by the whole salt water
  • Part (2): take the percent of the whole to get the salt
  • Part (3): run it backwards to recover the whole
  • Check all three against the definition