Ratios and Proportional Relationships

9.1: Ratios and Proportional Relationships. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 1 of 73 parts Practice Flashcards

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Allocate a total

Objective: Use a part-to-part ratio to divide a known total.

Prerequisite skills: ratio parts, equations, and unit rates. If two groups have ratio a:b, write their counts as ak and bk. Their total is (a+b)k, so divide the total by a+b to find one part. This lesson applies ratio structure to resource allocation. Common error: dividing the total by just one ratio term. Restrictions and interpretation: Maintain ratio order and distinguish proportional models from those with fixed offsets.

Formulas

first count = total·a/(a+b)

Worked examples

Try itA shipment of 650 books is divided between two branches in the ratio 23:3. How many books go to the first branch?Show answer

Let each ratio part be k books. There are 26 parts, so k=650/26=25. The first branch gets 23(25)=575 books. Therefore the requested answer is 575.

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Allocate a total

Objective: Use a part-to-part ratio to divide a known total.

Formulas

first count = total·a/(a+b)

Worked examples

Try itA shipment of 650 books is divided between two branches in the ratio 23:3. How many books go to the first branch?Show answer

Let each ratio part be k books. There are 26 parts, so k=650/26=25. The first branch gets 23(25)=575 books. Therefore the requested answer is 575.

Direct proportion

Objective: Recover the constant of proportionality and predict an output.

A direct proportion has the form y=kx with no fixed intercept. One nonzero input-output pair gives k=y/x. Use units to interpret k and apply the same ratio only under the stated constant conditions. Common error: adding the input difference to the output in a proportional model. Restrictions and interpretation: Maintain ratio order and distinguish proportional models from those with fixed offsets.

Formulas

$y=kx, k=y/x for x≠0$

Worked examples

Try itUnder a proportional recipe, 24 cups of grain require 96 cups of water. How many cups of water are needed for 7 cups of grain?Show answer

The constant water-to-grain ratio is 96/24=4. For 7 cups of grain, water=4×7=28 cups. Therefore the requested answer is 28.

Changing a ratio

Objective: Model a change to one category before imposing a new ratio.

Represent initial counts using a common ratio multiplier. After the stated addition or removal, write a new ratio equation with updated amounts. Do not apply the original ratio after the quantities change. Common error: changing both categories when only one receives an addition. Restrictions and interpretation: Maintain ratio order and distinguish proportional models from those with fixed offsets. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

a k/(b k + d)=new ratio

Worked examples

Try itA box has red and blue beads in the ratio 3:2. After 52 blue beads are added, the ratio is 3:4. How many red beads were there originally?Show answer

Original amounts are 3k and 2k. The new ratio requires 3k/(2k+52)=3/4. Cross-multiply: 12k=6k+156; thus k=26. Red beads=3k=78. Therefore the requested answer is 78.

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