Shanghai Maths · Second Semester · Chapter 4 companion

How to understand fraction sentences

Three common question patterns — and what each sentence is really asking.

Fraction of a whole Part–whole language Shanghai S2 · Ch4

Three patterns at a glance

These sentences use the same idea — part and whole — but they hide the unknown in different places. First, name what you know and what you need.

Sentence Example You are finding…
A. Fraction of a number $\dfrac{5}{7}$ of $35$ The part (how many)
B. What part of… What part of $15$ is $11$? The fraction
C. …of what number $18$ is $\dfrac{2}{3}$ of what number? The whole
Key idea: “Of” almost always means “out of this whole.” The whole is the number after “of,” unless the sentence asks for that whole.

A · Fraction of a number

Pattern A

$\dfrac{5}{7}$ of $35$

Means: take the whole $35$, cut it into $7$ equal parts, then take $5$ of those parts.

Whole (known)
$35$
How many equal parts
$7$ (denominator)
How many parts we take
$5$ (numerator)
Find
the part

Bar model

One whole $= 35$. Seven equal boxes. Five are shaded.

Each box $= 35 \div 7 = 5$. Five boxes $= 5 \times 5 = 25$.

Number sentence

$$\dfrac{5}{7} \text{ of } 35 = 35 \div 7 \times 5 = 25$$
$$\text{or}\quad \dfrac{5}{7} \times 35 = 25$$
Read it as: “Five sevenths of thirty-five” → divide by $7$, then multiply by $5$.

B · What part of … is …?

Pattern B

What part of $15$ is $11$?

Means: $11$ is how many fifteenths of the whole $15$? Write the relationship as a fraction.

Whole (known)
$15$ (after “of”)
Part (known)
$11$
Find
the fraction

How to hear the sentence

Reorder it in your mind:

$$11 \text{ is what part of } 15?$$

So:

$$\text{part} = \dfrac{11}{15}$$

Bar idea

The whole bar is $15$. The piece we care about is $11$. The fraction is “part over whole.”

$$\text{fraction} = \dfrac{\text{part}}{\text{whole}} = \dfrac{11}{15}$$
Watch the order: “What part of 15 is 11?” → whole $= 15$, part $= 11$ → $\dfrac{11}{15}$, not $\dfrac{15}{11}$.

C · … is a fraction of what number?

Pattern C

$18$ is $\dfrac{2}{3}$ of what number?

Means: $18$ is two of three equal parts of some whole. Find that whole.

Part (known)
$18$
Fraction of the whole
$\dfrac{2}{3}$
Find
the whole

Bar model

Three equal boxes make the whole. Two boxes together $= 18$.

One box $= 18 \div 2 = 9$. The whole has $3$ boxes $= 9 \times 3 = 27$.

Number sentence

$$\dfrac{2}{3} \text{ of } \square = 18$$
$$\square = 18 \div 2 \times 3 = 27$$
$$\text{or}\quad \square = 18 \div \dfrac{2}{3} = 18 \times \dfrac{3}{2} = 27$$
Read it as: “Eighteen is two-thirds of the whole” → first find one third ($18 \div 2$), then find three thirds ($\times 3$).

Compare side by side

A · $\dfrac{5}{7}$ of $35$ B · What part of $15$ is $11$? C · $18$ is $\dfrac{2}{3}$ of what?
Whole Known ($35$) Known ($15$) Unknown
Part Unknown Known ($11$) Known ($18$)
Fraction Known ($\dfrac{5}{7}$) Unknown Known ($\dfrac{2}{3}$)
Do $35 \div 7 \times 5$ $\dfrac{11}{15}$ $18 \div 2 \times 3$
Answer $25$ $\dfrac{11}{15}$ $27$

A reading habit before you calculate

  1. Underline “of.” The number after “of” is usually the whole — unless the sentence asks for that whole.
  2. Circle what is missing: the part, the fraction, or the whole?
  3. Say it in plain English: “Five sevenths of thirty-five,” “Eleven out of fifteen,” “Eighteen is two of three equal parts.”
  4. Draw equal boxes for the denominator, then shade the numerator.
  5. Check: Is your answer smaller than the whole when you took a proper fraction of it? (Type A.) Is your fraction less than $1$ when the part is less than the whole? (Type B.)

Quick try

  • $\dfrac{3}{5}$ of $40$ → Type A → $40 \div 5 \times 3 = 24$
  • What part of $20$ is $8$? → Type B → $\dfrac{8}{20} = \dfrac{2}{5}$
  • $24$ is $\dfrac{3}{4}$ of what number? → Type C → $24 \div 3 \times 4 = 32$

Key terms

Whole — the complete amount. Part — a piece of the whole. Numerator — how many equal parts we take. Denominator — how many equal parts make the whole. “Of” — points to the whole (or asks you to find it).